Verify valid probability distributions

A valid discrete probability distribution assigns each possible value of a random variable a probability from 0 to 1, with the probabilities across all mutually exclusive outcomes summing to exactly 1; this can be verified from a table, graph, or algebraic rule. The reasoning distinguishes invalid assignments, such as negative probabilities or totals other than 1, and supports interpreting expected outcomes and variability; continuous probability densities and more abstract distribution theory are not included.

Detailed Explanation: Verify valid probability distributions

For a discrete probability distribution, check two conditions:

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

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

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 $1$ .

Learn by doing: Verify valid probability distributions

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Probability Random Variables - Probability Table to Is Valid Table

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