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 μx

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) = Σxi · 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:

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How does the Expected Value Calculator work?
Free Expected Value Calculator - 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) = Σxi · 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) = ΣxI · 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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