[10 marks] Let X, Y, Z be random variables such that X is a standard...

 

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[10 marks] Let X, Y, Z be random variables such that ● X is a standard normal random variable, that is X~ conditional on X ~ N(0, 1) x, Y is a normal random variable with mean x and variance 1, Y~ N(x, 1) • conditional on X = x, and Y = y, Z is a normal random variable with mean x + y and variance 1, Z ~ N(x + y, 1) (a) Find the joint PDF of X, Y, Z. (b) Find E[X], E[Y], E[Z]. tor (X, Y, Z), that is [ Var(X) Cov(X,Y) Cov(X,Z) 1 tor (X, Y, Z), that is Cov(X,Y) Var(Y) Cov(Y,Z) Cov(X,Z) Cov(Y,Z) Var (Z) (c) Determine the following conditional probability density functions (PDFs): Find the covariance matrix of the random vec- • The conditional PDF of X given Y = y and Z = z. • The conditional joint PDF of X and Y given Z = z. (Do not present an integral as your answer.) [10 marks] Let X, Y, Z be random variables such that ● X is a standard normal random variable, that is X~ conditional on X ~ N(0, 1) x, Y is a normal random variable with mean x and variance 1, Y~ N(x, 1) • conditional on X = x, and Y = y, Z is a normal random variable with mean x + y and variance 1, Z ~ N(x + y, 1) (a) Find the joint PDF of X, Y, Z. (b) Find E[X], E[Y], E[Z]. tor (X, Y, Z), that is [ Var(X) Cov(X,Y) Cov(X,Z) 1 tor (X, Y, Z), that is Cov(X,Y) Var(Y) Cov(Y,Z) Cov(X,Z) Cov(Y,Z) Var (Z) (c) Determine the following conditional probability density functions (PDFs): Find the covariance matrix of the random vec- • The conditional PDF of X given Y = y and Z = z. • The conditional joint PDF of X and Y given Z = z. (Do not present an integral as your answer.)

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Given that x follows N01 Y Xx follows Nx 1 then fx y fY ... View the full answer