In this section we will introduce certain transformations of random variables for which the expected value of the transformation is the transformation of the expected value. We will also study variance of …

This lesson is concerned with linear combinations or if you would like linear transformations of the variables. Mathematically linear combinations can be expressed as shown in the expression below:

Thus, LLN also works for AR(1) data, thought they are not independent.

In some situations we are not interested in calculating probabilities and mean and variance for a specific random variable, but instead for a specific combination of random variables.

Nov 26, 2025 · Learn about the variance and mean of sums and combinations of random variables. This revision note covers the mean and variance of aX+b and aX+bY.

If the mean and variance of the original random variables are known, the mean and variance of the new random variables can be found from them using simple properties of random variables referred to …

Now, consider the random variable := + . First, note that is in fact a random variable. That is because both and. map from Ω to , meaning their sum will also map from Ω . Though that is all well and good, …

A major tool for data reduction & insight into why reject Ho regarding μs. Definition: A Linear Combination of p (random) variables X1, X2, · · · , Xp is a1X1 + a2X2 + · · · + apXp. Let X1. ′ X2 = …

Introduction So far we focus on one variable at a time. Examples. But, usually, we are interested in examining two or more r.v. together. Why? We need to use our measures of `co-movement'.

We can write where Being a linear transformation of a multivariate normal random vector, is also multivariate normal. Actually, it is univariate normal, because it is a scalar.