STAT 3023/3923
Semester 2 Statistical Inference 2021
Week 3 Tutorial
1. Let X and Y be two independent standard normal random variables.
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(a) Write down the joint density of the random vector X = (X, Y ).
(b) Let (R, θ) be the polar coordinates of the random vector X:
X = R cos θ, Y = R sin θ. Find the joint density of the random vector (R, θ).
(c) Find the marginal distributions of R and θ.
(d) Are R and θ independent?
(e) Find the density of R2.
2. Let X = (X, Y ) be a random vector with joint density f(x,y)=exp(−(x+y)), x>0, y>0.
(a) FindthejointdensityoftherandomvectorW=(X+Y,X−Y). (b) Find the marginal densities of X + Y and X − Y .
3. Consider the following joint probability function
(pλ)xe−pλ ((1 − p)λ)n−xe−(1−p)λ
P (X = x, N = n) = x! (n − x)! , where λ > 0 is a fixed parameter and x = 0,1,..,n : n = 0,1,…
(a) Show that N has a Poisson distribution with mean λ. (b) Find the conditional distribution of {X|N = n}.
(c) Find E(X|N = n).
4. If(X,Y)isuniformlydistributedonthetrianglex≥0, y≥0, x+y≤2findE(Y|X).
5. Let (X, Y ) have joint density
f(x, y) = 21x3e−x(1+y), x > 0, y > 0.
Find the conditional expectation of Y given X.
6. The random vector X = (X, Y ) has joint density
f(x,y)= 9(1+x+y) , x≥0, y≥0.
(b) Find E(Y |X). 7. Let
(a) Find a such that W = X1 + aX2 and X2 are independent.
(b) Hence, or otherwise, determine the conditional variance of X1 given X2 = 3.
(c) Evaluate P (X1 ≤ 2|X2 = 3).
2(1 + x)4(1 + y)4 (a) Find the marginal density of X.
X 3 3−1 X=1∼N2 , .
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