testseries

Question

U and V are two independent zero mean Gaussian random variables of variances 1/4 and 1/ 9. The probability P(3V  2U) is

Answer 1

Since U and V are given to be normal random variables, therefore their difference will also be a normal random variable.

Here, let Y=3V-2U, where Y is also a normal random variable with mean = 0 and variance = 3²  (1/9) + 2²  (1/4) = 2

So it will be symmetric about mean that is 0.

P(Y>=0)= 1/2 (by symmetry property)

Answer 2

Difference between two normal random variable is also a normal random variable.