Let XYZ be rvs such that X N01 and conditional on X x Y an

Let X,Y,Z be r.v.s such that X ~ N(0,1) and conditional on X = x, Y and Z are i.i.d. N(x,1).

i) Find the joint PDF of X,Y,Z.

ii) By definition, Y and Z are conditionally independent given X. Discuss intuitively whether or not Y and Z are also unconditionally independent.

iii) Find the joint PDF of Y and Z.

Solution

so, Y,Z | X ~ N(x,1)....X~ N(0,1).....

f [Y,Z | X=x] = f [ Y | X=x ] * f [ Z | X =x ] ..... = 1/ [2*pi]..

so, joint pdf of x,y,z = f ( x,y,z) = e^ (- x^2/2 ) / [ 2*pi ]...

ii ) the distributions of Y | X AND Z | X are both = 1/ [sqrt(2*pi) ] which does not depend on x..and joint pdf of x,y,z involves only terms of x so, if we remove x from this pdf by integrating, we will only have constants.so, Y and Z ARE independent unconditionally too!

iii) joint pdf of Y and Z = integration from -infinity to + infinity [  e^ (- x^2/2 ) / [ 2*pi ] ] = 1 / sqrt[2*pi]