Probability theory and statistics
Chapter 3 : Moments
By Lund University
A given random variables will have associated moments which can be calculated from its distribution functions and we look at moments in the first section. Once we have a random variable X, we can create a new one Y by a composition, Y = g(X) for some function and this is the topic of section 2. We conclude this chapter with the family of random variables called the normal random variables.
Expected value and variance
We have talked about discrete and continuous random variables and we have talked about distribution functions, from which we can calculate probabilities such as the probability that the random variable will take a value in a certain interval. In this section, we introduce moments of a random variable. A given random variable will have several different moments. The most important moments of a random variable is called the expected value. In this section, we will look at the definition of the expected value of a random variable, how it is calculated from the PDF or the PMF and how it is interpreted. We also look at the second most important moment of a random variable, the variance.
Expected value of a discrete random variable
Expected value of a continuous random variable
Problem: Expected value
The variance of a random variable
Function of a random variable and its moments
A common situation in probability theory is that we start with a given random variable X and we then define a new random variable Y as a function of the initial random variable X. It is in general possible to find the PDF or the PMF for Y if we know the PDF/PMF of X but this is typically a difficult problem. It turns out that we can often quite easily find the expected value of Y from the PDF/PMF of X without knowing the PDF/PMF of Y and we will look at precisely how this is done in this section. Finding the expected value and the variance of Y when Y is a linear function of X turns out to be particularly simple.
Function of a random variable
Expected value of a function of a random variable
Problem: Variance and functions of random variable
The expected value and variance of a linear function of a random variable
Problem: expected value and variance of a linear function of a random variable
The normal random variables
In this final section we looked at an entire family of random variables called the normal random variables to which the standard normal random variable belongs. Given any expected value and any positive variance, this family contains a random variable with this particular expected value and variance. The family has many other useful properties which we will look at later on in the course.