Expected Value

Expected Value Definition:

An Expected Value is the weighted average of a random variable. Also called the mean.

Expected Value Notation

We represent the the expected value of a random variable X as E(X) or μ_x

Expected Value Example

Take a fair coin where the probability of flipping a head is 1/2 and the probability of flipping a tail is 1/2.
Expected Number of Heads in 2 coin flips = E(H)
E(H) = 2 flips * 1/2 probability of heads
E(H) = 1

Discrete Random Variable where P(x) is the probability mass function of X:

E(X) = Σx_i · P(x)

Continuous Random Variable

E(X) = _-∞∫^∞x · P(x)dx
where x is the value of the continuous random variable X and P(x) is the probability density function

Expected Values of a constant times a random variable

When a is a constant and X and Y are random variables, we have
E(aX) = aE(x)
E(X + Y) = E(X) + E(Y)

Expected Value of a Constant:

Given a constant c, we have
E(c) = c

Expected Values of a product

When X and Y are independent random variables, we have
E(X · Y) = E(X) · E(Y)

Variance Definition Using Expected Value:

Variance of a random variable is the average value of the square distance from the mean value. In other words, how close the random variable is distributed near the mean value.
σ² = Var(X) = E(X - μ)²

Expected Values Calculator:

To see more on Expected Value, you can ask check out our Expected Value Search

How does the Expected Value Calculator work?

This lesson walks you through what expected value is, expected value notation, the expected value of a discrete random variable, the expected value of a continuous random variable, and expected value properties.

What 7 formulas are used for the Expected Value Calculator?

E(aX) = aE(x)
E(X + Y) = E(X) + E(Y)
E(c) = c
When X and Y are independent random variables, we have E(X � Y) = E(X) · E(Y)
σ² = Var(X) = E(X - μ)²
Discrete Random Variable: E(X) = Σx_i · P(x)
Continuous Random Variable: E(X) = _-∞∫^∞x · P(x)dx

For more math formulas, check out our Formula Dossier

What 6 concepts are covered in the Expected Value Calculator?

constant: a value that always assumes the same value independent of how its parameters are varied
continuous random variable: one which takes an infinite number of possible values
discrete random variable: a type of variable whose value depends upon the numerical outcomes of a certain random phenomenon
expected value: predicted value of a variable or event
E(X) = Σx_I · P(x)
product: The answer when two or more values are multiplied together
statistics: Statistics is a discipline concerned with the analysis of data and decision making based upon data.

Expected Value Calculator Video

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