Linear combinations of normal random variables. One property that makes the normal distribution extremely tractable from an analytical viewpoint is its closure under linear combinations: the linear combination of two independent random variables having a normal distribution also has a normal distribution.
What makes the normal distribution tractable?
One property that makes the normal distribution extremely tractable from an analytical viewpoint is its closure under linear combinations: the linear combination of two independent random variables having a normal distribution also has a normal distribution.
What is linear combinations?
Linear Combinations is the answer! More importantly, these properties will allow us to deal with expectations (mean) and variances in terms of other parameters and are valid for both discrete and continuous random variables. Let’s quickly review a theorem that helps to set the stage for the remaining properties.
How do you find the mean and variance of a linear combination?
Suppose X 1, X 2, …, X n are n independent random variables with means μ 1, μ 2, ⋯, μ n and variances σ 1 2, σ 2 2, ⋯, σ n 2. Then, the mean and variance of the linear combination Y = ∑ i = 1 n a i X i, where a 1, a 2, …, a n are real constants are:
What is the sum of more than two independent normal random variables?
The sum of more than two independent normal random variables also has a normal distribution, as shown in the following example. Example Let be mutually independent normal random variables, having means and variances . Then, the random variable defined ashas a normal distribution with mean and variance.
How do you calculate the expected value of a linear combination?
For example, if we let X represent the number that occurs when a blue die is tossed and Y, the number that happens when an orange die is tossed. This means we can determine their respective probability distributions and expected values and use it to calculate the expected value of the linear combination 3X – Y of the random variables X and Y:
