Question

The random variable W = X – 3Y + Z + 2 where X, Y and Z are three independent Normal random variables, with E[X]=E[Y]=E[Z]=2 and Var[X]=9,Var[Y]=1,Var[Z]=3.

The pdf of W is:

		Uniform
		Poisson
		Binomial
		Normal
		None of the other pdfs.

Answer 1

Here, X,Y,Z are independent

The new random variable variable is W

W=X-3Y+Z+2

[ since, X,Y,Z are independent]

We know the linearity property of normal distribution.

i.e, linear function of independent normal random variable

follow normal distribution.

So, the random variable W follow normal distribution with mean 0 and variance 21.

So, the pdf of W is a Normal.