Linear Combinations of Random Variables

The joint distribution of a particular pair of linear combinations of random variables which are independent of each other is a bivariate normal distribution. It forms the basis for all calculations involving arbitrary means and variances relating to the more general bivariate normal distribution. The property of rotational symmetry implies that the joint distribution of any two linear combinations of the two variables is bivariate normal.

The same concept can also be extended to linear combinations of any number of independent normal variables. By applying scaling the condition the two variables need not be standard in nature.

If there are two linear combinations of independent normal variables V and W then their joint distribution is bivariate normal. The paramaters of the bivariate distribution can then be easily calculated.

The following result is implied from the above definitions –

Two linear combinations of independent normal variables are independent only when the two sets of data are not correlated with each other.

What can be said about linear combinations of random variables can also be said about functions of random varaibles that are formed by a linear relationship. A function of random variables is itself a random variable and it can either be built from linear or non-linear relationships between variables.

Given that Y is a linear combination of variables _X_1, _X_2,…,Xp and constants _c_1, _c_2, …, cp

Y= _c_1_X_1 + _c_2_X_2 + … + cpXp

Multi-Variate Normal Distribution

This content is for paid members only.

Join our membership for lifelong unlimited access to all our data science learning content and resources.