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

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