Mean and SD of Random Variables

The long-run average

• The mean (or expected value 期望值) of a random variable is its long-run average outcome.
• Written $\mu_X$ or $E(X)$ — the balance point of the distribution.
• It's a weighted average: each value weighted by its probability.
• Over many repetitions, the average result closes in on $\mu_X$.

Computing the mean

• $$\mu_X = \sum x_i \, P(x_i)$$
• Multiply each value by its probability, then add them all up.
• The mean need not be a possible value (a "family of $2.5$ kids").
• It's an average, not a prediction of any single trial.

A discrete probability distribution with its mean and standard deviation

Variance and standard deviation

• The variance 方差 $\sigma_X^2 = \sum (x_i - \mu_X)^2 \, P(x_i)$ — the probability-weighted average squared distance from the mean.
• The standard deviation $\sigma_X = \sqrt{\sigma_X^2}$ brings it back to the original units.
• Bigger $\sigma_X$ = outcomes are more spread out around the mean.
• Square the deviations, weight by probability, sum, then square-root.

Interpreting the SD

• $\sigma_X$ is the typical distance of an outcome from the mean $\mu_X$.
• Small $\sigma_X$: results cluster tightly near the expected value.
• Large $\sigma_X$: results swing widely from trial to trial.
• Together, $\mu_X$ (center) and $\sigma_X$ (spread) summarize the whole distribution.

Write from an expected value of 3

• A random variable pays 0 with probability 0.25 and 4 with probability 0.75.
• The lesson expected value 2 is another distribution.