Skill: Verify valid probability distributions

For a discrete probability distribution, check two conditions:

1. Every probability is between $0$ and $1$, inclusive:

$$0 \leq P(X=x) \leq 1$$

2. The probabilities of all possible outcomes add to exactly $1$:

$$\sum P(X=x)=1$$

If both conditions are true, the distribution is valid.

Worked example

Suppose the random variable $X$ represents the number of books a student reads in a month.

$x$	$0$	$1$	$2$	$3$
$P(X=x)$	$0.20$	$0.35$	$0.30$	$0.15$

Step 1: Check each probability.

The probabilities are $0.20$, $0.35$, $0.30$, and $0.15$. Each one is between $0$ and $1$, so the first condition is satisfied.

Step 2: Add the probabilities.

$$0.20+0.35+0.30+0.15=1.00$$

The total is exactly $1$, so the second condition is also satisfied.

Conclusion: This is a valid probability distribution because every probability is between $0$ and $1$, and the probabilities sum to $1$.