5.4 Discrete random variables

A discrete random variable takes separate values with probabilities p(x)≥0 and Σp(x)=1. Its probability table defines the sample space and supports expectation calculations.

Check the support and total probability before using formulas. A probability mass at one value is not a continuous density over an interval.

For X=0,1,2 with probabilities 0.2,0.5,0.3, the probabilities sum to 1 and describe every possible outcome.

A discrete variable can have many values; “discrete” means countable separated outcomes, not necessarily only two.

If independent trials have constant success probability p, the number X of the first success has P(X=r)=(1−p)^{r−1}p for r=1,2,… .

Confirm that trials are independent and p is constant. “First success” counts trials including the successful one; a waiting-time variant may count failures instead.

With p=0.2, P(X=4)=0.8³×0.2=0.1024.

A geometric model does not describe the number of successes in a fixed number of trials; that is binomial.

For discrete X, E(X)=Σxp(x), E(X²)=Σx²p(x), and Var(X)=E(X²)−[E(X)]². Standard deviation is the square root of variance.

Variance is measured in squared units; standard deviation returns to the original units. For aX+b, E=aE(X)+b and Var=a²Var(X).

If E(X)=4 and Var(X)=9, then Var(2X+1)=36 and E(2X+1)=9.

Adding a constant changes the mean but not the variance; multiplying by a changes variance by a², not a.