Revision sheet

Discrete distributions

• Values with probabilities in $[0, 1]$ summing to 1.
• Add probabilities for events.

Watch out: values are not automatically equally likely.

Expected value and SD

$E(X) = \sum x P(x)$; $\text{SD} = \sqrt{\sum (x - \mu)^2 P(x)}$.

Watch out: weight by probabilities; include losses.

Combining random variables

• Means add (and subtract).
• Independent: variances add, for sums and differences.

Watch out: never add SDs directly.

Binomial probabilities

$P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}$ in a binomial setting.

Watch out: multiply by the number of arrangements.

Binomial mean and SD

$\mu = np$, $\sigma = \sqrt{np(1 - p)}$.

Watch out: the SD is the square root of $np(1 - p)$.

Geometric distribution

$P(X = k) = (1 - p)^{k - 1}p$, $\mu = 1/p$.

Watch out: $k - 1$ failures before the first success.