Section 10.1

Fair Prices from the Definition

Expected payoffs computed straight from the distribution, with no decision to make: chuck-a-luck, roulette bets, paying per box searched, St Petersburg, heavy-tailed variables.

Toolkit

These problems compute $\mathbb{E}[X]=\sum_x x\,\mathbb{P}(X=x)$ (or $\int x f(x)\,dx$ ) directly from an explicit distribution, with no optimisation involved. Linearity lets you split payoffs into pieces and sum their expectations separately.

Key tools:

Indicator/linearity trick: for a sum of payouts or positions, write $X=\sum_i X_i$ and use $\mathbb{E}[X]=\sum_i \mathbb{E}[X_i]$ , even when the $X_i$ are dependent.
Geometric/negative-binomial tails: if $\mathbb{P}(N=k)=(1-p)^{k-1}p$ , then $\mathbb{E}[N]=1/p$ ; sums like $\sum_k k\,r^k$ via $\sum kr^k=\frac{r}{(1-r)^2}$ .
Binomial weighting: for $Y\sim\mathrm{Bin}(n,p)$ , $\mathbb{E}[g(Y)]=\sum_k g(k)\binom{n}{k}p^k(1-p)^{n-k}$ .
Uniform-position symmetry: if an object’s rank among $n$ equally likely orderings is uniform on $\{1,\dots,n\}$ , its expected rank is $(n+1)/2$ .
Divergence check: a series $\sum_n p_n v_n$ with constant terms (as in doubling payoff against halving probability) diverges, giving infinite expectation.
Breakeven condition: expected value is invariant under a modification when the modification’s own expected multiplicative/additive effect equals $1$ or $0$ .

Problems