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 of a Discrete Random Variable
E(X) = Σxi · ƒ(x)
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 a coin flip = E(H) E(H) = H * P(H) + T * P(T) E(H) = 1 * 0.5 + 0 * 0.5 E(H) = 0.5 + 0 E(H) = 0.5 or 1/2
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