Question

Suppose X, Y, and Z are independent Gaussian random variables with σx = 0.2, σy = 0.3, σz = 1, μx=3.0, μy=7.7, and μz = 0 Determine the following:

(a) The joint PDF of X, Y, and Z

(b) Pr( (7.6<Y<7.8) | (X<3, Z<0)

c) The Correlation matrix and covariance matrix of X, Y, and Z

Answer 1

(a) -

It is given that X, Y, Z are independent random variables. Hence, joint pdf is the product of individual pdfs.

(b) -

We know if two random variables are independent then P(X|Y) = P(X)

Hence,

We know that follows standard normal distribution.

Hence,

(c) -

As X, Y, Z are independent, they are uncorrelated. And hence all the covariances and correlations between X,Y,Z are 0.

Hence, all the off diagonal entries are zero.

Diagonal entries of covariance matrix are variances of X,Y,and Z.

Diagonal entries of correlation matrix are correlations of X,Y,and Z with itself. That is they are all 1.

Covariance matrix =

Correlation matrix =

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