Contents
How to find the conditional distribution of Y given X?
To find the conditional distribution of Y given X = x, assuming that (1) Y follows a normal distribution, (2) E ( Y | x), the conditional mean of Y given x is linear in x, and (3) Var ( Y | x), the conditional variance of Y given x is constant. To learn how to calculate conditional probabilities using the resulting conditional distribution.
What are the assumptions for a normal distribution?
That’s what we’ll do in this lesson, that is, after first making a few assumptions. First, we’ll assume that (1) Y follows a normal distribution, (2) E ( Y | x), the conditional mean of Y given x is linear in x, and (3) Var ( Y | x), the conditional variance of Y given x is constant.
Which is the conditional variance of a normal distribution?
If the conditional distribution of Y given X = x follows a normal distribution with mean μ Y + ρ σ Y σ X ( x − μ X) and constant variance σ Y | X 2, then the conditional variance is: Because Y is a continuous random variable, we need to use the definition of the conditional variance of Y given X = x for continuous random variables.
Which is the correct equation for a bivariate normal distribution?
After pulling the conditional variance through the integral on the left side of the equation, and rewriting the right side of the equation as an expectation, we have: Now, by the definition of a valid p.d.f., the integral on the left side of the equation equals 1:
When do you use a conditional density function?
Density functions determine continuous distributions. If a continuous distri-bution is calculated conditionally on some information, then the density is called a conditional density. When the conditioning information involves another random variable with a continuous distribution, the conditional den-
How to find the joint distribution of X and Y?
To find the joint distribution of X and Y assuming that (1) X follows a normal distribution, (2) Y follows a normal distribution, (3) E ( Y | x), the conditional mean of Y given x is linear in x, and (4) Var ( Y | x), the conditional variance of Y given x is constant. To learn the formal definition of the bivariate normal distribution.
What does it mean when Y follows a normal distribution?
The continuous random variable Y follows a normal distribution for each x. The conditional mean of Y given x, that is, E ( Y | x), is linear in x. Recall that that means, based on our work in the previous lesson, that: The conditional variance of Y given x, that is, Var ( Y | x) = σ Y | X 2 is constant, that is, the same for each x.