How to Find the Expected Value of a Random Variable?

A step-by-step guide to the expected value of a random variable

• \(EV\): the expected value
• \(P(X)\): the probability of the occurrence of the event
• \(n\): the number of repetitions of the event

• \(EV\): the expected value
• \(P(X_i)\): the probability of the event
• \(X_i\): the event

The Expected Value of a Random Variable, Example 1:

To find the expected value, use this formula: \(\color{blue}{EV=\sum P(X_i) \times X_i}\)

\(EV=0 (\frac{1}{6})+ 1 (\frac{1}{6}) + 2 (\frac{1}{6})+ 3 (\frac{1}{6})\)

\(=0+\frac{1}{6}+\frac{2}{6}+\frac{3}{6}\) \(=\frac{0+1+2+3}{6}=\frac{6}{6}=1\)

\(EV=1\)

The Expected Value of a Random Variable, Example 2:

When you roll a die, you get paid \(\(2\) for an odd number and \(\) 1\) for an even number. Find the expected value of money you get for one roll of the die.

The sample space of the experiment is:\((1,2,3,4,5,6)\)

First, we draw the table of the probability distribution for a single roll of a die and the amount paid for each result.

Then, to find the expected value, use this formula: \(\color{blue}{EV=\sum P(X_i) \times X_i}\)

\(EV= 2 (\frac{1}{6})+ 1 (\frac{1}{6})+ 2 (\frac{1}{6}) + 1 (\frac{1}{6})+ 2 (\frac{1}{6})+ 1 (\frac{1}{6})\)

\(=\frac{2}{6}+\frac{1}{6}+\frac{2}{6}+\frac{1}{6}+\frac{2}{6}+\frac{1}{6}\) \(=\frac{2+1+2+1+2+1}{6}\)=\(\frac{9}{6}\)

\(EV=\frac{9}{6}=1.5\)

Exercises for the Expected Value of a Random Variable

• A men’s soccer team plays zero football, one or two days a week. The probability that they will play on day zero is \(0.2\), the probability that they will play one day is \(0.5\) and the probability that they will play two days is \(0.4\). Find the expected value of the number of days per week the men’s soccer team plays football.

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• A hospital researcher is interested in the number of times a typical patient calls the nurse after surgery during a \(12\)-hour shift. The following information was obtained from a random sample of \(40\) people. What is the expected value?

• In a city, \(12\%\) of families have three children, \(50\%\) of families have two children, \(22\%\) of families have one child, and \(11\%\) of families have no children. What is the expected value of children in a family?

• \(\color{blue}{1.3}\)
• \(\color{blue}{\frac{29}{10}}\)
• \(\color{blue}{1.58}\)

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Frequently Asked Questions

How do you find the expected value of a discrete random variable?

Multiply each possible value by its probability, then add the results: E[X] = Σ x · P(X = x). If X takes the values 1, 2 and 3 with probabilities 0.2, 0.5 and 0.3, then E[X] = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1.0 + 0.9 = 2.1. First check that the probabilities add up to 1.

How do you find the expected value of a continuous random variable?

Replace the sum with an integral: E[X] = ∫ x · f(x) dx over the whole range, where f(x) is the probability density function. For X uniform on [0, 4], f(x) = 1/4, so E[X] = ∫₀⁴ x/4 dx = 16/8 = 2. That matches the midpoint, as you would expect for a uniform distribution.

How do you find the expected value of two random variables?

For a sum, just add the expected values: E[X + Y] = E[X] + E[Y]. This works even if X and Y are not independent. Rolling two fair dice, each die has an expected value of 3.5, so the expected total is 3.5 + 3.5 = 7. Constants work the same way: E[2X + 5] = 2E[X] + 5.

How do you find the expected value of two random variables multiplied?

If X and Y are independent, E[XY] = E[X] · E[Y]. For two fair dice, the expected product is 3.5 × 3.5 = 12.25. If they are not independent, this shortcut fails, and you must use the joint distribution: multiply each pair of values by the probability of that pair and add them all up.

How do you find the expected value of a binomial random variable?

For a binomial random variable with n trials and success probability p, the expected value is E[X] = np. If a student guesses on 20 multiple-choice questions with 4 choices each, p = 1/4, so the expected number correct is 20 × 1/4 = 5. You get the same answer by adding the expected value of each trial.

How do you compute the expected value and variance of a random variable?

Find E[X] first, then use Var(X) = E[X²] − (E[X])². For one fair die, E[X] = 3.5 and E[X²] = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6. So Var(X) = 91/6 − 12.25 = 35/12 ≈ 2.92, and the standard deviation is about 1.71.

Related Lessons You May Like

• How to find the probability of an event
• How to solve probability problems
• How to find the measures of central tendency
• How to find experimental probability
• Probability distribution

For a friendly introduction to probability and statistics that builds the intuition before the formulas, Pre-Algebra for Beginners covers the prerequisites and includes a probability chapter. For deeper distributions, expected value, and combinatorics, Algebra II for Beginners goes further.