Random Variables

Probability Distribution

A discrete random variable lists each possible value with its probability.

The probabilities must add to 1.

Example

Check that a discrete distribution's probabilities sum to one.

highlighted = computed this step

Step 1 — Distribution table

List each outcome and its probability.

xP114212314\begin{array}{cc}\text{x} & \text{P} \\ 1 & \hlmath{\frac{1}{4}} \\ 2 & \hlmath{\frac{1}{2}} \\ 3 & \hlmath{\frac{1}{4}}\end{array}

Step 2 — First probabilities

Add the first two probabilities.

14+12=34\frac{1}{4} + \frac{1}{2} = \hlmath{\frac{3}{4}}

Step 3 — Normalization

Add the last probability to check the total.

34+14=1\frac{3}{4} + \frac{1}{4} = \hl{1}

Step 4 — Valid distribution

The probabilities sum to one.

∑P(x)=1\sum P(x)= \hl{1}
probability-distribution For a valid discrete distribution, sum all probabilities to get 1.