# Finding joint probability distributions from marginal distributions

Question:

I was solving test papers where I found this one.

My doubt:

I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But my problem is that, here Y | Z~ N(1+Z, 1). What I cannot understand is how the mean of the given normal distribution is (1+Z) as Z is itself a random variable ? Or does this denote something different ?

Also I cannot understand how to solve part "C".

Thank you.

Consider in this way: You and I will go to draw random numbers from normal distribution. The process is I draw one random number $$Z_1$$ from $$N(0,1)$$ first, then based on what I get $$z_1$$, you draw a random number $$Y_1$$ from $$N(z_1+1,1)$$. Repeat this process $$N$$ times, we get $$(Z_i, Y_i)$$, $$i=1,...,N$$, where $$Z_i\sim N(0,1)$$ and $$Y_i|Z_i\sim N(Z_i+1,1)$$.
C: From B, we know that $$U = 1+Z$$. So $$U=1.7 ==> Z = 0.7$$. $$E(Y|U=1.7) = E(Y|Z=0.7)$$ $$Y|Z \sim N(1+Z,1) ==> E(Y|Z) = 1+Z$$
• Conditional on $z$, then $z$ should be constant. Commented Nov 24, 2018 at 23:06