Question

Given that the conditional distribution of Y, denoted as (Y∣X=x), follows a normal distribution with mean x and variance x², and that X follows a uniform distribution on the interval [0, 1], find: (a) The expected value E(Y) and (b) The variance Var(Y).

Answer 1

The expected value E(Y) is 1/3 and the variance Var(Y) is 1/4.

Step-by-step explanation:

Expected Value:

To find the expected value E(Y), we need to integrate the product of the probability density function of X and the conditional mean of Y given X. Since X follows a uniform distribution on the interval [0, 1], the probability density function of X is 1. Therefore, the expected value E(Y) is:

E(Y) = ∫(0 to 1) x * x dx

E(Y) = ∫(0 to 1) x^2 dx

E(Y) = [x^3/3] from 0 to 1

E(Y) = 1/3 - 0/3

E(Y) = 1/3

Variance:

To find the variance Var(Y), we need to integrate the product of the probability density function of X, the conditional variance of Y given X, and the conditional mean squared of Y given X. Since X follows a uniform distribution on the interval [0, 1], the probability density function of X is 1. Therefore, the variance Var(Y) is:

Var(Y) = ∫(0 to 1) x * x^2 dx

Var(Y) = ∫(0 to 1) x^3 dx

Var(Y) = [x^4/4] from 0 to 1

Var(Y) = 1/4 - 0/4

Var(Y) = 1/4