Mean of Discrete Random Variable

Statistics • Discrete Random Variables and Their Probability Distributions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

Mean of a Discrete Random Variable

Enter a discrete probability distribution and compute the mean (expected value) using the method: multiply each value by its probability, then sum the products.

Distribution input

Input mode Probabilities Frequencies

Value of x	Probability P(x)	Row

1) Validity check + mean (expected value)

ΣP(x) = — Condition 1: — Condition 2: — Valid distribution: — μ = E(x) = —

x	P(x)	x · P(x)	Notes
The mean is the sum of the x · P(x) column: μ = Σ[x · P(x)].

Interpretation.

—

2) Visualizations

Bar graph of P(x) with the mean marked

Each bar height equals P(x). The vertical line shows μ (expected value).

Contribution bars for x · P(x)

The sum of these contributions equals μ.

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Mean (expected value) of a discrete random variable

The mean of a discrete random variable x, denoted by μ, is the average value expected in the long run if the experiment is repeated many times.

Formula.

μ = Σ [ x · P(x) ]

The mean is also called the expected value:

E(x) = Σ [ x · P(x) ]

Probability distribution conditions

1) For each value of x: 0 ≤ P(x) ≤ 1

2) ΣP(x) = 1

A mean is computed from a valid probability distribution.

Example

Multiply each value by its probability and add the products.

x	P(x)	x · P(x)	Sum
0	0.15	0 × 0.15 = 0.00	μ = 1.80
1	0.20	1 × 0.20 = 0.20
2	0.35	2 × 0.35 = 0.70
3	0.30	3 × 0.30 = 0.90

Therefore: μ = 0.00 + 0.20 + 0.70 + 0.90 = 1.80. This does not mean the outcome must be 1.80; it means the long-run average is expected to be about 1.80 per repetition.

Frequently Asked Questions

What is the mean (expected value) of a discrete random variable?

It is the long-run average outcome you expect if the random process is repeated many times. It is computed from the probability distribution using mu = sum( x * P(x) ).

How do I calculate the mean from a probability distribution table?

Multiply each x value by its probability P(x), then add all the products. The calculator shows this in the x * P(x) column and sums it to get mu.

What makes a discrete probability distribution valid?

Every probability must be between 0 and 1, and the probabilities across all outcomes must add up to 1. A mean is computed from a valid distribution.

Can I use frequencies instead of probabilities to compute the mean?

Yes. When you enter frequencies, the calculator converts them to probabilities by dividing each frequency by the total frequency, then computes the mean from the resulting P(x) values.

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Distribution input

1) Validity check + mean (expected value)

2) Visualizations

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