 # 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 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:

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?

1. E(aX) = aE(x)
2. E(X + Y) = E(X) + E(Y)
3. E(c) = c
4. When X and Y are independent random variables, we have E(X � Y) = E(X) · E(Y)
5. σ2 = Var(X) = E(X - μ)2
6. Discrete Random Variable: E(X) = Σxi · P(x)
7. 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.