What Is Mean of Random Variable?
The mean of a random variable is the average of its possible values, each weighted by its probability. It is also called the expected value, E[X] or μ.
Enter the values of X and their probabilities, which must add to 1. The calculator multiplies and adds, shows the table, and also gives the mean of aX + b using the rule E[aX + b] = a·E[X] + b.
The mean is the long-run average over many repeats. It does not have to be a value that X can actually take.
How to Use the Mean of Random Variable Calculator
- Enter the possible values of X.
- Enter the matching probabilities. They must add to 1.
- Read E[X], and set a and b if you need the mean of a linear change.
Mean of Random Variable Formula
Mean
Linear change
Mean of Random Variable Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| X | The random variable | |
| P(x) | Probability of the value x | All must add to 1 |
| E[X] | Expected value, the mean | Σ x·P(x) |
| a, b | Constants in Y = aX + b | E[Y] = a·E[X] + b |
Mean of Random Variable Worked Example
With the example values loaded in the calculator above, the steps are:
- The values add to probability 1, so this is a valid distribution.
- E[X] = Σ x · P(x) = 0×0.1 + 1×0.3 + 2×0.4 + 3×0.2 = 1.7.
- Linear rule: E[aX + b] = a·E[X] + b = 2 × 1.7 + 1 = 4.4.
Result: Mean E[X] = 1.7; E[2X + 1] = 4.4.
Worked example
Let X take 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2. Then E[X] = 0 × 0.1 + 1 × 0.3 + 2 × 0.4 + 3 × 0.2 = 1.7.
For Y = 2X + 1 the mean is 2 × 1.7 + 1 = 4.4.
What the mean tells you
If you repeat the experiment many times, the average of the results gets close to E[X]. Any single result can be far from it.
Where Mean of Random Variable Is Used
Probability homework
Find the expected value from a probability table.
Games and decisions
Work out the average payoff of a game.
Linear transforms
Find the mean after scaling or shifting a variable.
Common Mean of Random Variable Mistakes to Avoid
- Check the probabilities add to 1 first.
- Use the linear rule instead of recomputing for Y = aX + b.
Mean of Random Variable Terms and Definitions
- Random variable
- A variable whose value depends on chance.
- Expected value
- The probability-weighted average of the possible values.
Mean of Random Variable FAQs
How do you find the mean of a random variable?
Multiply each value by its probability and add the products.
Is the mean the same as the expected value?
Yes. For a random variable they are the same thing.
What is the mean of aX + b?
a times the mean of X, plus b.
Can the mean be a value X cannot take?
Yes. A fair die has a mean of 3.5 but never shows 3.5.