2-Expectation  and  variance  of  continuous  random variables  

We have defined the expected value in Unite 2 of a discrete random variable X by

If X is a continuous random variable having probability density function f (x), then,

it is easy to see that the analogous definition is to define the expected value of X by

Find E(X) when the density function of X is 

If X is a continuous random variable with probability density function  f (x), then, for any real-valued function g, 

The density function of X is given by 

If a and b are constants, then

𝐸(𝑎𝑋+𝑏)=𝑎𝐸(𝑋)+𝑏

The proof of Corollaryis the same as the one given for a discrete random variable.

The only modification is that the sum is replaced by an integral and the probability mass

function by a probability density function. The variance of a continuous random variable is defined exactly 

as it is for a discrete random variable, namely,

if X is a random variable with expected value μ, then the variance of X is defined (for any type of random variable) by 

Find Var(X) for X as given in Example 4. 

We first compute 

It can be shown that, for constants a and b, 

The proof mimics the one given for discrete random variables.