The Coin Game That Makes Everyone Poorer

What to remember

• Markov: for $X \ge 0$, $P(X \ge a) \le E[X]/a$. It is sharp: mass at $0$ and at $a$ attains it, and it fails for variables that can go negative.
• Chebyshev is Markov applied to $(X - \mu)^2$: $P(|X - \mu| \ge k\sigma) \le 1/k^2$, so at most 25% of any distribution lies $2\sigma$ or more from the mean (4.55% for a normal).
• Applied to a sample mean, Chebyshev gives $P(|\bar X - \mu| \ge \varepsilon) \le \sigma^2/(n\varepsilon^2)$, a one-line proof of the weak LLN and a guaranteed sample size (2000 flips for $\pm 0.05$ at 95%, against 385 from the CLT).
• Jensen: $E[f(X)] \ge f(E[X])$ for convex $f$, reversed for concave $f$, with gap $\approx \tfrac12 f''(\mu)\,\sigma^2$. So $E[X^2] \ge (E[X])^2$, $E[1/X] \ge 1/E[X]$ and $E[\ln X] \le \ln E[X]$.
• Compounded growth runs near $\mu - \sigma^2/2$ (volatility drag): $+20\%$ or $-10\%$ averages 5% but compounds at about 3.9% a year.
• Convex payoffs gain from spread: a call is worth more than its payoff at the expected price, which is why option value rises with volatility and a long option is long gamma.
• In a multiplicative bet the mean grows at the arithmetic mean of the factors and the typical path at the geometric mean (AM–GM is Jensen on $\ln$): $\times 1.5$ or $\times 0.6$ averages $+5\%$ a flip, yet the median loses 5.1% a flip, and staking the Kelly fraction $\frac14$ turns it into a gain of 0.62%.