Expected value of a random variable: Definition: Let X be a continuous random variable with range [a, b] and probability density function f(x). The expected value of X is defined by E(X) = of (x) dr. Let's see how this compares with the formula for a discrete random variable: E(X) = >IP(x.). i=1 f(x) is often written as p(x) and it denotes the probability density function. A probability density function is a function f defined on an interval (a, b) and having the following properties. (a) f(x) 2 0 for every x (1 ) ( f(x) dx = 1 Expected value is also called the mean or average.Some properties of E(X) 1. If X and Y are random variables on a sample space ? then E(X + Y) = E(X) + E(Y) 2. If a and b are constants then E(ax + b) = QE(X) + b.5. The standard normal function is defined as: Z = X - P ~ N(0, 1). The expected value of Z i.e. E(Z) is 0. Using the properties above, find E(X)