Random variables
Expectation and variance, the binomial formula, the Poisson distribution and the number of trials until success, the normal law, the three-sigma rule and the central limit theorem.
How to solve it
A random variable is a number that depends on chance: the score on a die, the number of hits, the height of a person picked at random. Its two main characteristics are the expectation (the average over a long series) and the variance (how widely the values spread around it).
Step by step
- Expectation is linear: $\mathbb E(aX + b) = a\,\mathbb E X + b$. Variance: $\operatorname{Var}(aX + b) = a^2\operatorname{Var} X$ — a shift does not affect it.
- Exactly $k$ successes in $n$ independent trials: the binomial formula.
- Rare events with a known mean count $\lambda$: Poisson, $P(k) = \frac{\lambda^k}{k!}e^{-\lambda}$. The mean number of trials until the first success is $\frac{1}{p}$.
- The normal law: convert a value into standard deviations $z = \frac{x - \mu}{\sigma}$ and use the rule: within $\pm 1\sigma$ lie $68\%$, within $\pm 2\sigma$ $95\%$, within $\pm 3\sigma$ $99.7\%$.
Common mistakes
- Shifting the variance along with the expectation: $\operatorname{Var}(3X - 5) = 9\operatorname{Var} X$, and the $-5$ plays no part.
- Forgetting the binomial coefficient — the number of ways to choose which trials succeed.
- Confusing the standard deviation with the variance: $\sigma = \sqrt{\operatorname{Var} X}$.
- For one tail of the normal law taking everything outside the band instead of half.