---
title: "Continuous Random Variables"
book: "High School Mathematics"
subject: math
language: en
chapter: 35
exercises: 9
source: https://one-course.com/books/math/2/en/chapter/35-continuous-random-variables
---

# Chapter 35 — Continuous Random Variables

Waiting times, physical measurements, proportions: many random quantities take a continuum of values, and no single value has positive [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution). Probabilities are then computed by integrating a *[density](#def-g12-contdist-density)*. This chapter studies the uniform, exponential and [normal distributions](#def-g12-contdist-normal), and uses the normal law to quantify the fluctuations of polls and [samples](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-sample).

## 35.1 Probability densities

**Definition 35.1 (Density, continuous random variable).**

A *probability density* on an [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval) $I$ is a [continuous](https://one-course.com/books/math/2/en/chapter/21-limits-and-continuity#def-g12-limcont-continuity), nonnegative [function](https://one-course.com/books/math/2/en/chapter/11-functions-and-variations#def-g11-func-function) $f$ on $I$ with $\int_I f(t)\,\dd t = 1$ (the [integral](https://one-course.com/books/math/2/en/chapter/25-integration#def-g12-integ-area) over an unbounded $I$ being understood as a limit of [integrals](https://one-course.com/books/math/2/en/chapter/25-integration#def-g12-integ-area) over growing bounded [intervals](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval)). A [random variable](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-rv) $X$ has density $f$ if for all $a \leq b$ in $I$:

$$
\P(a \leq X \leq b) = \int_a^b f(t)\,\dd t .
$$

Probabilities are *areas under the [density](#def-g12-contdist-density) curve*. In particular $\P(X = a) = \int_a^a f = 0$ for every single value $a$: only [intervals](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval) carry [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution), and $\P(a \leq X \leq b) = \P(a < X < b)$.

**Definition 35.2 (Expectation and variance).**

For $X$ with [density](#def-g12-contdist-density) $f$ on $I$:

$$
\E(X) = \int_I t\,f(t)\,\dd t, \qquad
\V(X) = \int_I \bigl(t - \E(X)\bigr)^2 f(t)\,\dd t
= \E(X^2) - \E(X)^2 .
$$

The rules of Chapters 14–15 (linearity of [expectation](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp), additivity of [variance](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp) for [independent](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#def-g12-sums-indep) variables, Bienaymé–Chebyshev) remain valid; their proofs, with [integrals](https://one-course.com/books/math/2/en/chapter/25-integration#def-g12-integ-area) in place of sums, are admitted at this level.

## 35.2 Uniform distribution

**Definition 35.3 (Uniform distribution).**

$X$ follows the *uniform distribution* $\mathcal U\intcc ab$ if its [density](#def-g12-contdist-density) is constant, $f(t) = \frac{1}{b-a}$ on $\intcc{a}{b}$. Then for $\intcc{c}{d} \subseteq \intcc ab$, $\P(c \leq X \leq d) = \frac{d - c}{b - a}$: [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) proportional to length.

**Proposition 35.4.**

If $X \sim \mathcal U\intcc ab$, then $\E(X) = \dfrac{a+b}{2}$ and $\V(X) = \dfrac{(b-a)^2}{12}$.

**Proof.** $\E(X) = \int_a^b \frac{t}{b-a}\dd t
= \frac{b^2 - a^2}{2(b-a)} = \frac{a+b}{2}$. For the [variance](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp), $\E(X^2) = \frac{b^3 - a^3}{3(b-a)} = \frac{a^2 + ab + b^2}{3}$, and

$$
\V(X) = \frac{a^2 + ab + b^2}{3} - \frac{(a+b)^2}{4}
= \frac{4a^2 + 4ab + 4b^2 - 3a^2 - 6ab - 3b^2}{12}
= \frac{(b - a)^2}{12}. \qedhere
$$

∎

## 35.3 Exponential distribution

**Definition 35.5 (Exponential distribution).**

For $\lambda > 0$, $X$ follows the *exponential distribution* $\mathcal E(\lambda)$ if its [density](#def-g12-contdist-density) on $\intco{0}{+\infty}$ is

$$
f(t) = \lambda\,\eu^{-\lambda t}.
$$

Then $\P(X \leq x) = 1 - \eu^{-\lambda x}$ and $\P(X > x) = \eu^{-\lambda x}$ for $x \geq 0$.

**Proposition 35.6.**

If $X \sim \mathcal E(\lambda)$: $\E(X) = \dfrac{1}{\lambda}$ and $\V(X) = \dfrac{1}{\lambda^2}$.

**Proof.** Integrating by parts on $\intcc{0}{A}$:

$$
\int_0^A t\,\lambda\eu^{-\lambda t}\dd t
= \bigl[-t\,\eu^{-\lambda t}\bigr]_0^A + \int_0^A \eu^{-\lambda t}\dd t
= -A\eu^{-\lambda A} + \frac{1 - \eu^{-\lambda A}}{\lambda}
\xrightarrow[A\to+\infty]{} \frac1\lambda,
$$

using $A\eu^{-\lambda A} \to 0$ ([Theorem 23.4](https://one-course.com/books/math/2/en/chapter/23-exponential-and-logarithm#thm-g12-exp-growth)). A second [integration by parts](https://one-course.com/books/math/2/en/chapter/25-integration#thm-g12-integ-ibp) gives $\E(X^2) = \frac{2}{\lambda^2}$, whence $\V(X) = \frac{2}{\lambda^2} - \frac{1}{\lambda^2}$. ∎

**Theorem 35.7 (Memorylessness).**

If $X \sim \mathcal E(\lambda)$, then for all $s, t \geq 0$:

$$
\pcond{X > s}{X > s + t} = \P(X > t) .
$$

The [exponential distribution](#def-g12-contdist-expo) is the law of lifetimes *without aging* (radioactive nuclei, not light bulbs).

**Proof.**

$$
\pcond{X > s}{X > s+t}
= \frac{\P(X > s + t)}{\P(X > s)}
= \frac{\eu^{-\lambda(s+t)}}{\eu^{-\lambda s}} = \eu^{-\lambda t}
= \P(X > t). \qedhere
$$

∎

![The exponential density - x (here = 1): the tail area beyond t (red) is - t, and memorylessness says every tail looks like the whole distribution rescaled.](https://one-course.com/images/onecourse/chapters/math-2/g12-contdist/fig-60c0c3684efd.svg)

*The exponential [density](#def-g12-contdist-density) $\lambda\eu^{-\lambda x}$ (here $\lambda = 1$): the tail area beyond $t$ (red) is $\eu^{-\lambda t}$, and memorylessness says every tail looks like the whole [distribution](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-rv) rescaled.*

## 35.4 Normal distribution

**Definition 35.8 (Normal distribution).**

$X$ follows the *standard normal distribution* $\mathcal N(0, 1)$ if its [density](#def-g12-contdist-density) on $\R$ is

$$
\varphi(t) = \frac{1}{\sqrt{2\pi}}\,\eu^{-t^2/2}
$$

(the bell curve). More generally, $X \sim \mathcal N(\mu, \sigma^2)$ if $\dfrac{X - \mu}{\sigma} \sim
\mathcal N(0,1)$; then $\E(X) = \mu$ and $\V(X) = \sigma^2$.

![The normal density: about 68.3\% of the mass lies within of the mean, 95.4\% within 2, 99.7\% within 3.](https://one-course.com/images/onecourse/chapters/math-2/g12-contdist/fig-4494c7446eb1.svg)

*The normal [density](#def-g12-contdist-density): about $68.3\%$ of the mass lies within $\sigma$ of the [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean), $95.4\%$ within $2\sigma$, $99.7\%$ within $3\sigma$.*

**Remark 35.9.**

The factor $\frac{1}{\sqrt{2\pi}}$ makes the total area $1$ — a famous computation (the *Gaussian [integral](https://one-course.com/books/math/2/en/chapter/25-integration#def-g12-integ-area)*) done at university. There is no elementary formula for $\int \varphi$: normal probabilities are read from tables or calculators.

**Theorem 35.10 (De Moivre–Laplace, admitted).**

Let $X_n \sim \mathcal B(n, p)$ and $Z_n = \dfrac{X_n - np}{\sqrt{np(1-p)}}$ (the standardized binomial). Then for all $a \leq b$,

$$
\P(a \leq Z_n \leq b) \xrightarrow[n\to+\infty]{}
\int_a^b \varphi(t)\,\dd t .
$$

*This result is admitted at this level.* It explains the universal appearance of the bell curve: a binomial with large $n$ is approximately normal — and (central limit theorem, university) so is any sum of many small [independent](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#def-g12-sums-indep) effects.

**Method 35.11 (Fluctuation interval at 95%95\%95%).**

Since a normal variable falls within $1.96$ [standard deviations](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-variance) of its [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) with [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) $0.95$, for $n$ large a [frequency](https://one-course.com/books/math/2/en/chapter/8-descriptive-statistics#def-g10-stats-series) $F_n = \frac{X_n}{n}$ of a $\mathcal B(n,p)$ [sample](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-sample) satisfies

$$
\P\left(p - 1.96\sqrt{\tfrac{p(1-p)}{n}} \leq F_n \leq
p + 1.96\sqrt{\tfrac{p(1-p)}{n}}\right) \approx 0.95 .
$$

- *Fluctuation [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval)* (testing): if a claimed $p$ puts the observed [frequency](https://one-course.com/books/math/2/en/chapter/8-descriptive-statistics#def-g10-stats-series) outside this [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval) , reject the claim at the $5\%$ level.
- *[Confidence interval](#met-g12-contdist-fluctuation)* (estimating): the simplified [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval) $\left[F_n - \frac{1}{\sqrt n},\ F_n + \frac{1}{\sqrt n}\right]$ contains $p$ with [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) at least $0.95$ (using $1.96\sqrt{p(1-p)} \leq 1$ , see [Exercise 34.8](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#exo-g12-sums-8) ).

**Example 35.12.**

A poll of $n = 1000$ voters gives a candidate $52\%$. The [confidence interval](#met-g12-contdist-fluctuation) $52\% \pm \frac{1}{\sqrt{1000}} \approx 52\% \pm 3.2\%$ still contains $50\%$: the poll alone does not establish that the candidate is ahead.

## 35.5 Exercises

**Exercise 35.1 ★.**

A bus passes every $15$ minutes; a traveler arrives at a uniformly random time. Let $X \sim \mathcal U\intcc{0}{15}$ be the waiting time. Compute $\P(X \leq 5)$, $\P(4 \leq X \leq 10)$, $\E(X)$ and $\sigma(X)$.

**Solution of Exercise 35.1.**

$\P(X \leq 5) = \frac{5}{15} = \frac13$; $\P(4 \leq X \leq 10) = \frac{6}{15} = \frac25$; $\E(X) = \frac{15}{2} = 7.5$ min; $\sigma(X) = \frac{15}{\sqrt{12}} = \frac{5\sqrt3}{2} \approx 4.3$ min.

**Exercise 35.2 ★.**

Check that $f(t) = \frac{3}{8}t^2$ is a [density](#def-g12-contdist-density) on $\intcc{0}{2}$, and compute $\E(X)$ and $\P(X \geq 1)$ for a variable $X$ with this [density](#def-g12-contdist-density).

**Solution of Exercise 35.2.**

$f \geq 0$ and $\int_0^2 \frac38 t^2\,\dd t = \frac38\left[\frac{t^3}{3}\right]_0^2
= \frac38 \times \frac83 = 1$: a [density](#def-g12-contdist-density).

$$
\E(X) = \int_0^2 \frac38 t^3\,\dd t = \frac38 \times \frac{16}{4} = \frac32,
\qquad
\P(X \geq 1) = \int_1^2 \frac38 t^2\,\dd t = \frac{8 - 1}{8} = \frac78 .
$$

**Exercise 35.3 ★.**

The lifetime (in years) of an electronic component follows $\mathcal E(0.2)$.

1. Compute the expected lifetime and $\P(X > 5)$ .
2. The component has already worked $3$ years. What is the [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) it works at least $5$ more?

**Solution of Exercise 35.3.**

*1.* $\E(X) = \frac{1}{0.2} = 5$ years; $\P(X > 5) = \eu^{-0.2\times5} = \eu^{-1} \approx 0.37$.

*2.* By memorylessness ([Theorem 35.7](#thm-g12-contdist-memoryless)), $\pcond{X > 3}{X > 8} = \P(X > 5) = \eu^{-1} \approx 0.37$: the three years of service change nothing.

**Exercise 35.4 ★★.**

The half-life of a radioactive atom whose lifetime follows $\mathcal E(\lambda)$ is the [median](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-median) $m$: $\P(X > m) = \frac12$. Express $m$ as a [function](https://one-course.com/books/math/2/en/chapter/11-functions-and-variations#def-g11-func-function) of $\lambda$ and compare with the [expectation](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp). Which is larger, and why does that make sense for a skewed [distribution](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-rv)?

**Solution of Exercise 35.4.**

$\P(X > m) = \eu^{-\lambda m} = \frac12$ gives $m = \frac{\ln 2}{\lambda} \approx \frac{0.69}{\lambda}$, smaller than $\E(X) = \frac1\lambda$. The exponential [density](#def-g12-contdist-density) has a long right tail: a few atypically long lifetimes pull the *[mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean)* above the *[median](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-median)*, the value that half the population exceeds. (This is the half-life of [Example 23.9](https://one-course.com/books/math/2/en/chapter/23-exponential-and-logarithm#ex-g12-exp-halflife): half the atoms survive it, even though the average lifetime is longer.)

**Exercise 35.5 ★★.**

Heights in a population follow $\mathcal N(175,\ 7^2)$ (in cm). Using the $68$–$95$–$99.7$ rule, estimate the proportion of the population with height between $168$ and $182$ cm, above $189$ cm, and below $154$ cm.

**Solution of Exercise 35.5.**

$168 = \mu - \sigma$ and $182 = \mu + \sigma$: about $68\%$. $189 = \mu + 2\sigma$: above it lies half of the remaining $100 - 95.4 = 4.6\%$, so about $2.3\%$. $154 = \mu - 3\sigma$: about $\frac{100 - 99.7}{2} = 0.15\%$.

**Exercise 35.6 ★★.**

A machine fills bags labeled $500$ g; the mass filled follows $\mathcal N(\mu,\ 4^2)$. Regulations demand that at most $2.5\%$ of bags weigh less than $500$ g. Using the $95\%$ rule, what minimal setting of $\mu$ complies?

**Solution of Exercise 35.6.**

$\P(X < 500) \leq 2.5\%$ means $500$ must sit at least two [standard deviations](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-variance) below the [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) (the normal leaves $2.3\% \approx 2.5\%$ below $\mu - 2\sigma$; with the conventional $1.96$, the reasoning is identical):

$$
\mu - 2\sigma \geq 500 \iff \mu \geq 500 + 8 = 508 \text{ g}.
$$

The machine must be set to $508$ g on average — the price of the guarantee is $8$ g of “free” product per bag.

**Exercise 35.7 ★★.**

A die claimed to be fair is rolled $1200$ times and shows a six $260$ times (as in [Exercise 34.7](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#exo-g12-sums-7)).

1. Compute the $95\%$ fluctuation [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval) for the [frequency](https://one-course.com/books/math/2/en/chapter/8-descriptive-statistics#def-g10-stats-series) of sixes of a fair die over $1200$ rolls.
2. Is the observed [frequency](https://one-course.com/books/math/2/en/chapter/8-descriptive-statistics#def-g10-stats-series) inside? Compare the strength of this conclusion with the Bienaymé–Chebyshev analysis.

**Solution of Exercise 35.7.**

*1.* With $p = \frac16$, $n = 1200$: $\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{5/36}{1200}} \approx 0.0108$, so the $95\%$ fluctuation [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval) is

$$
\frac16 \pm 1.96 \times 0.0108 \approx \intcc{0.146}{0.188}.
$$

*2.* The observed [frequency](https://one-course.com/books/math/2/en/chapter/8-descriptive-statistics#def-g10-stats-series) $\frac{260}{1200} \approx 0.217$ lies far outside: the fairness hypothesis is rejected at the $5\%$ level. The deviation is about $4.6$ [standard deviations](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-variance) — for a normal [approximation](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-approx), a [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) of order $10^{-6}$, far more conclusive than the $\leq 4.6\%$ bound from Bienaymé–Chebyshev ([Exercise 34.7](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#exo-g12-sums-7)).

**Exercise 35.8 ★★★.**

Before an election, a poll of $n$ people will estimate the score $p$ of a candidate by the observed [frequency](https://one-course.com/books/math/2/en/chapter/8-descriptive-statistics#def-g10-stats-series) $F_n$.

1. With the [confidence interval](#met-g12-contdist-fluctuation) of [Method 35.11](#met-g12-contdist-fluctuation) , what [sample](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-sample) size guarantees a margin of $\pm 2$ points?
2. The candidates are separated by $1$ point in reality. Explain why no realistic poll can reliably call the winner, however well conducted.

**Solution of Exercise 35.8.**

*1.* Margin $\frac{1}{\sqrt n} \leq 0.02$ requires $n \geq 2500$ people.

*2.* To distinguish scores $1$ point apart, the margin must be well under $0.5$ point, requiring $n \geq \left(\frac{1}{0.005}\right)^2 =
40\,000$ by the simplified formula — already impractical for most polls. Worse, the statistical margin only accounts for *sampling* error; systematic biases (unrepresentative [samples](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-sample), non-response, last-minute swings) do not shrink as $n$ grows and typically exceed one point. A $1$-point race is genuinely too close to call.

**Exercise 35.9 ★★★.**

Let $X$ have [density](#def-g12-contdist-density) $f$ on $\intcc ab$ and let $Y = \alpha X + \beta$ with $\alpha > 0$.

1. Show that $\P(c \leq Y \leq d)  = \P\left(\frac{c - \beta}{\alpha} \leq X \leq  \frac{d-\beta}{\alpha}\right)$ , and deduce that $Y$ has [density](#def-g12-contdist-density) $g(y) = \frac{1}{\alpha} f\left(\frac{y - \beta}{\alpha}\right)$ .
2. Deduce that if $Z \sim \mathcal N(0,1)$ , then $\mu + \sigma Z$ has [density](#def-g12-contdist-density) $\frac{1}{\sigma\sqrt{2\pi}}\,  \eu^{-(y - \mu)^2/(2\sigma^2)}$ — the general normal [density](#def-g12-contdist-density) .

**Solution of Exercise 35.9.**

*1.* Since $\alpha > 0$, $c \leq \alpha X + \beta \leq d
\iff \frac{c - \beta}{\alpha} \leq X \leq \frac{d - \beta}{\alpha}$, so

$$
\P(c \leq Y \leq d)
= \int_{(c-\beta)/\alpha}^{(d-\beta)/\alpha} f(t)\,\dd t .
$$

The substitution $y = \alpha t + \beta$ (i.e. reading the area in the $y$ variable, $t = \frac{y - \beta}{\alpha}$, $\dd t = \frac{\dd y}{\alpha}$) turns this into $\int_c^d \frac{1}{\alpha} f\left(\frac{y-\beta}{\alpha}\right)\dd y$: the [function](https://one-course.com/books/math/2/en/chapter/11-functions-and-variations#def-g11-func-function) $g(y) = \frac1\alpha f\left(\frac{y-\beta}{\alpha}\right)$ is a [density](#def-g12-contdist-density) of $Y$.

*2.* With $f = \varphi$, $\alpha = \sigma$, $\beta = \mu$:

$$
g(y) = \frac{1}{\sigma}\,\varphi\!\left(\frac{y - \mu}{\sigma}\right)
= \frac{1}{\sigma\sqrt{2\pi}}\,
\exp\left(-\frac{(y-\mu)^2}{2\sigma^2}\right),
$$

which is therefore the [density](#def-g12-contdist-density) of $\mathcal N(\mu, \sigma^2) = \mu + \sigma\,\mathcal N(0,1)$.

## 35.6 Problem: Waiting for the bus, measuring the crowd

**Problem 35.1.**

Weekend problem — three densities run the world: total ignorance, memoryless waiting, and the bell of many small causes; plus the paradox that makes your bus always late

Why does your bus always seem to take longer than the timetable promises? Why is the class an average student sits in bigger than the average class? Why do your friends have more friends than you? One sly piece of mathematics — length-biased sampling — answers all three, and it lives in this chapter, between the uniform, exponential and [normal distributions](#def-g12-contdist-normal) ([Definition 35.3](#def-g12-contdist-uniform), [Definition 35.5](#def-g12-contdist-expo), [Definition 35.8](#def-g12-contdist-normal)). This final problem of the volume puts the three densities to work and ends where the whole series points: the bell curve.

**Part I — Three characters.**

1. A bus arrives at a uniformly random time in the next $30$ minutes. Compute $\P(\text{wait} \leq 10)$ , the expected wait, and $\sigma$ ( [Proposition 35.4](#prop-g12-contdist-uniformmoments) ).
2. Check that $f(x) = 2x$ on $\intcc{0}{1}$ is a [density](#def-g12-contdist-density) , and compute $\E(X)$ and $\P\left(X > \frac12\right)$ .
3. Phone calls reach a helpline as a memoryless stream: the wait for the next call is exponential with [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $10$ minutes. Compute $\P(T > 15)$ and the [median](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-median) wait — why is the [median](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-median) *less* than the [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) ?
4. You have already waited $20$ minutes. What is $\P(T > 35 \mid T > 20)$ ( [Theorem 35.7](#thm-g12-contdist-memoryless) )? Interpret in one sentence.
5. Match each wait to its right model — uniform, exponential, or neither: (a) the click of a Geiger counter; (b) a metro running every $8$ minutes, your arrival unsynchronized; (c) the next call at the helpline; (d) the failure of a light bulb that *wears out* . Justify (d) with the memorylessness theorem.

**Part II — The bell at work.** (Use the normal landmarks: $68\,\%$ within $\sigma$, $95\,\%$ within $2\sigma$, $99.7\,\%$ within $3\sigma$.)

6. Adult heights follow roughly $\mathcal N(172,\ 8)$ (cm). What fractions lie in $\intcc{164}{180}$ , $\intcc{156}{188}$ , $\intcc{148}{196}$ ?
7. Estimate the proportion taller than $190$ cm ( [z-score](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#pb-g11-stat-1) first; table or calculator).
8. IQ scores are calibrated to $\mathcal N(100, 15)$ . What fraction scores above $130$ — and roughly one person in how many?
9. A machine cuts bolts with lengths $\mathcal N(50,\ 0.1)$ mm and the specification is $\intcc{49.8}{50.2}$ . What share is rejected? Industry celebrates “six sigma” processes, whose specifications sit at $\pm 6\sigma$ : what does that buy, and why do factories chase it?
10. De Moivre–Laplace ( [Theorem 35.10](#thm-g12-contdist-tcl) ): for $100$ fair coin flips, approximate $\P(45 \leq \text{heads} \leq 55)$ ( [continuity](https://one-course.com/books/math/2/en/chapter/21-limits-and-continuity#def-g12-limcont-continuity) : $\pm 0.5$ ; $\sigma = 5$ ). Which wooden machine of [Problem 19.1](https://one-course.com/books/math/2/en/chapter/19-the-binomial-distribution#pb-g11-binom-1) does this theorem smooth into a curve?
11. From [Exercise 35.8](#exo-g12-contdist-8) : a $\pm 2$ -point margin needs about $n = 2\,400$ . Compute the [sample](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-sample) needed for $\pm 0.5$ points — and conclude, in one sentence, why a race separated by one point cannot be honestly called by any poll.

**Part III — The bus paradox.**

12. Buses arrive as a memoryless stream with [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) gap $10$ minutes. You reach the stop at an arbitrary moment. Intuition says the expected wait is $5$ minutes (half a gap). What does the memorylessness theorem say instead?
13. Diagnose the intuition with a toy timetable: gaps alternate $5$ and $15$ minutes ( [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) gap $10$ ). Compute the [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) that a uniformly random arrival lands in a long gap, then the true expected wait — and name the culprit: arrivals [sample](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-sample) gaps with [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) proportional to their *length* .
14. In the memoryless stream, the gap you land in has expected length $20$ minutes — twice the typical gap (the time back to the previous bus and forward to the next are *both* exponential with [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $10$ ). Reconcile this with question 12 and state the *inspection paradox* in one sentence.
15. Same sting on land: a university runs nine classes of $10$ students and one of $110$ . Compute the average class size, then the class size experienced by the *average student* . Which number will the brochure quote, and which is the students’ lived truth?
16. The friendship paradox — “your friends have, on average, more friends than you” — is the same mathematics. Say in one sentence what plays the role of the long gap.

**Part IV — Closing the volume.**

17. Simulation, the practitioners’ bridge: if $U$ is uniform on $\intoo{0}{1}$ , show that $T = -10\ln U$ is exponential with [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $10$ (compute $\P(T > t)$ ). Every random-waiting simulation in industry runs on this one-line trick.
18. An old game programmers’ recipe builds a bell: add twelve [independent](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#def-g12-sums-indep) uniforms on $\intcc{0}{1}$ and subtract $6$ . Give the [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) and [variance](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp) of the sum, and explain — Galton board in hand — why the result is nearly normal.
19. Markets crash harder than the bell allows: the 1987 crash was described as a “ $25\sigma$ [event](https://one-course.com/books/math/2/en/chapter/18-probability-and-random-variables#def-g11-prob-model) ”, which under normality has [probability](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-distribution) around $10^{-137}$ . What is the correct conclusion — about the world, or about the model? (One sentence, the statistician’s catechism.)
20. Finale, and farewell to the volume: the three characters in one line each — uniform (ignorance within bounds), exponential (hazard without memory), normal (the sum of many small causes); the inspection paradox as the chapter’s sting; and the arc of the whole series — from counting cartons in grade 6 to the bell curve that measures every crowd — with the university volumes waiting where the limits, the [integrals](https://one-course.com/books/math/2/en/chapter/25-integration#def-g12-integ-area) and the central limit theorem take over.

**Solution of Problem 35.1.**

**1.** $\P(\text{wait} \leq 10) = \frac{10}{30} =
\frac13$; $\E = 15$ min; $\sigma = \frac{30}{\sqrt{12}} \approx 8.7$ min.

**2.** $\int_0^1 2x\,\dd x = 1$: a [density](#def-g12-contdist-density). $\E(X) = \int_0^1 2x^2\,\dd x = \frac23$; $\P\left(X > \frac12\right) = \int_{1/2}^1 2x\,\dd x =
1 - \frac14 = \frac34$.

**3.** $\P(T > 15) = \eu^{-15/10} \approx 0.22$. [Median](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-median): $\eu^{-m/10} = \frac12$: $m = 10\ln 2 \approx 6.9$ min — less than the [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $10$ because the exponential’s long right tail (rare huge waits) drags the [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) upward while leaving the [median](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-median) at the “typical” wait.

**4.** By memorylessness, $\P(T > 35 \mid T > 20) = \P(T > 15) \approx 0.22$: the twenty minutes already served buy nothing — the stream does not age.

**5.** (a) exponential (radioactive decay is the model memorylessness was invented for); (b) uniform on $\intcc{0}{8}$ (a schedule with a random offset); (c) exponential; (d) neither: a worn bulb is *more* likely to fail in the next hour than a new one, so $\P(T > s + t \mid T > s) < \P(T > t)$: aging contradicts the memorylessness theorem.

**6.** $68\,\%$, $95\,\%$, $99.7\,\%$: the three landmark [intervals](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval).

**7.** $z = \frac{190 - 172}{8} = 2.25$: upper tail $\approx 1.2\,\%$.

**8.** $z = 2$: about $2.3\,\%$ — roughly one person in $44$.

**9.** The spec sits at $\pm 2\sigma$: about $95.4\,\%$ pass, $4.6\,\%$ rejected — ruinous at scale. At $\pm 6\sigma$ the failure rate drops to about two parts per *billion*: six-sigma buys the right to mass-produce without mass-inspecting.

**10.** $\sigma = 5$; with [continuity](https://one-course.com/books/math/2/en/chapter/21-limits-and-continuity#def-g12-limcont-continuity) correction, $\P \approx \P\left(\abs Z \leq \frac{5.5}{5}\right) =
\P(\abs Z \leq 1.1) \approx 0.73$. The theorem smooths the Galton board of [Problem 19.1](https://one-course.com/books/math/2/en/chapter/19-the-binomial-distribution#pb-g11-binom-1): the staircase of slots becomes the [continuous](https://one-course.com/books/math/2/en/chapter/21-limits-and-continuity#def-g12-limcont-continuity) bell.

**11.** $n \approx \frac{1.96^2 \times 0.25}{0.005^2}
\approx 38\,400$ people — and at that size the *non*-sampling errors (frame, refusals, lies) dwarf the margin: a one-point race is beyond honest polling, which is why serious institutes then say “too close to call”.

**12.** Memorylessness: from the moment you arrive, the remaining wait is exponential with [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $10$ — the full $10$ minutes, not $5$. The timetable’s “[mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) gap $10$” misleads: your wait has the same [distribution](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-rv) as a whole gap.

**13.** Random arrivals land proportionally to gap length: $\P(\text{long gap}) = \frac{15}{20} = \frac34$. Expected wait: $\frac34 \times \frac{15}{2} + \frac14 \times
\frac52 = 5.625 + 0.625 = 6.25$ min — more than the naive $5$, though the [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) gap is $10$. Culprit: *length-biased sampling* — long gaps catch more passengers.

**14.** Backward and forward waits are each exponential with [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $10$ (memorylessness runs in both directions from your arrival): the gap containing you averages $20$ minutes — twice the typical gap. Inspection paradox in one sentence: *the [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval) you happen to inspect is not a typical [interval](https://one-course.com/books/math/2/en/chapter/1-numbers-and-sets-of-numbers#def-g10-numbers-interval), because you were more likely to fall into a big one.*

**15.** Average class: $\frac{9 \times 10 + 110}{10} =
20$ students. Student-experienced average: $\frac{90 \times 10 + 110 \times 110}{200} = \frac{13\,000}
{200} = 65$ students. The brochure prints $20$; three in five students sit in the class of $110$ and live the $65$.

**16.** Popular people appear on many friend-lists, so sampling “a friend” is length-biased towards the sociable — your friends are drawn from the long gaps of the social timetable.

**17.** For $t \geq 0$: $\P(T > t) = \P(-10\ln U > t) = \P\left(U <
\eu^{-t/10}\right) = \eu^{-t/10}$: exactly the exponential’s survival [function](https://one-course.com/books/math/2/en/chapter/11-functions-and-variations#def-g11-func-function) — one [logarithm](https://one-course.com/books/math/2/en/chapter/23-exponential-and-logarithm#def-g12-exp-ln) converts the computer’s uniform noise into any waiting time.

**18.** Each uniform has [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $\frac12$ and [variance](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp) $\frac{1}{12}$: the sum of twelve has [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $6$ and [variance](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp) $1$ — so the recipe outputs [mean](https://one-course.com/books/math/2/en/chapter/17-descriptive-statistics#def-g11-stat-mean) $0$, [variance](https://one-course.com/books/math/2/en/chapter/33-random-variables-and-the-binomial-distribution#def-g12-randvar-exp) $1$. Twelve [independent](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#def-g12-sums-indep) small pushes summed: the Galton mechanism, and by De Moivre–Laplace’s blessing the histogram is already bell-shaped to the eye.

**19.** About the model: markets are not sums of many *[independent](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#def-g12-sums-indep)* small causes — panics correlate everything (the 2008 lesson of the previous problem) and produce fat tails no normal [density](#def-g12-contdist-density) owns. When the data whispers $10^{-137}$, the statistician’s catechism answers: *the model is rejected, not the day*.

**20.** Uniform: all positions equal, the honest “I know only the bounds”. Exponential: the hazard that never ages — decays, arrivals, clicks. Normal: the democratic bell of many small [independent](https://one-course.com/books/math/2/en/chapter/34-sums-of-random-variables-and-the-law-of-large-numbers#def-g12-sums-indep) causes — heights, errors, averages. Sting: what you [sample](https://one-course.com/books/math/2/en/chapter/9-probability-and-sampling#def-g10-proba-sample) is biased by how you bumped into it — buses, classes, friends. And the arc: the child who counted juice cartons in grade 6 has, ten volumes of problems later, measured the crowd with a curve; the epsilon, the [integral](https://one-course.com/books/math/2/en/chapter/25-integration#def-g12-integ-area) and the central limit theorem are waiting in the university volumes to explain *why* the bell tolls for everything.