What is Expectation?

Example 18.10

For the game of Example 18.6:

On average the player loses about $17$ cents per game: the game is unfavorable. A game is called fair when the expected gain is $0$.

Example 18.15

For the game of Example 18.6: $\E(G^2) = \frac46 \times 4 + \frac16 \times 9 + \frac16 \times 16 = \frac{16 + 9 + 16}{6} = \frac{41}{6}$, so

The loss of $0.17$ per game on average comes with swings of typical size $2.6$ euros — expectation without standard deviation tells only half the story.

Example 22.10 (Transfer in action)

For $X \sim \mathcal P(\lambda)$, compute $\E\bigl(\frac1{1+X}\bigr)$ — the law of $\frac1{1+X}$ itself is awkward, but transfer never asks for it:

Two lessons. Computationally: recognizing a shifted exponential series is the whole work — transfer reduces expectations of $f(X)$ to series manipulation. Structurally: the naive plug-in value would be $\frac1{1 + \E X} = \frac1{1 + \lambda}$, and the true answer is larger,

exactly as Jensen’s inequality for the convex function $t \mapsto \frac1{1+t}$ demands. Expectations of convex images sit above the naive plug-in value, and transfer plus a series check makes the abstract inequality concrete.

Example 22.12 (Products, with and without independence)

Roll two fair dice. If $Y$ is the second die (independent of the first), $\E(XY) = \E(X)\E(Y) = 3.5^2 = 12.25$. If instead $Y = X$ (the “product” of a die with itself),

same marginal laws in both scenarios, different joint laws, different product expectations. The moral, worth engraving: $\E(XY)$ is a functional of the pair, not of the two marginals — and the gap $\E(X^2) - \E(X)^2 \approx 2.92$ is, by König–Huygens, precisely the variance $\frac{35}{12}$ of the die.