Review Problems for the Final
Question 1. Suppose that X is uniformly distributed over (0, 1). Let Y = − log X. Find
both the cdf and pdf of Y , and then find its expected value.
Question 2. Suppose that X has a Cauchy distribution; that is X has the pdf
f (x) =
1
,
π(1 + x2 )
−∞ < x < ∞.
Find the median of X.
Question 3. Let X have the standard normal pdf
1
2
fX (x) = √ e−x /2 ,
2π
−∞ < x < ∞.
(a) Find the pdf of Y = X 2 and calculate EY .
(b) Find the pdf of Y = |X| and find its mean and variance.
Question 4. Find the moment generating function corresponding to the pdf
f (x) =
1 −|x−α|/β
e
,
2β
−∞ < x < ∞,
−∞ < α < ∞,
β > 0.
Question 5. Let (X, Y ) be uniformly distributed on the unit square. (Note that this is
equivalent to assuming that X and Y are independent Uniform(0, 1))
(a) Find the pdf fZ of Z = X + Y .
(b) Find the pdf fU of U = max(X, Y ).
(c) Find the pdf fV of V = min(X, Y ).
Question 6. Let X and Y be i.i.d. Exponential(1) rvs and set Z = X + Y .
(a) Find the pdf fZ of Z = X + Y .
(b) Find the pdf fU of U = max(X, Y ).
(c) Find the pdf fV of V = min(X, Y ).
(d) Find the pdf fW of W = |X − Y |.
1
X
.
X +Y
Hint: |X − Y | ≡ max(X, Y ) − min(X, Y ).
(e) Find the pdf of fR of R =
Question 7. Let X be a continuous rv and Y be a discrete rv. Prove that
f (y|x)fX (x) = f (x|y)fY (y).
Question 8. Let the joint pdf of (X, Y ) be given by
(
2, if 0 < x < 1, 0 < y < x
f (x, y) =
0, Otherwise.
(a) Find the distribution of Y |X.
(b) Find E(Y |X).
(c) Find Cov(X, Y ).
Question 9. Suppose that the joint distribution of (X, Y ) is specified by the conditional
distribution of X|Y and the marginal distribution of Y (≡ ”p”) as follows:
X|Y ∼binomial(n, Y ),
Y ∼uniform(0, 1),
(discrete)
(continuous)
Find E(Y |X).
Question 10. Let X ∼ gamma(α, λ) and Y ∼ gamma(β, λ) be independent gamma
random variables with the same scale parameter and define
U = X + Y,
W =
X
.
X +Y
Find the joint pdf of (U, W ), find the marginal pdfs of U and W , and show that U and W
are independent.
Question 11. Let (X, Y ) be a continuous bivariate random vector with joint pdf fX,Y (x, y)
on R2 . Find the joint pdf fR,Θ (r, θ) of (R, Θ), where (X, Y ) → (R, Θ) is the 1-1 transformation whose inverse is given by
X = R cos Θ, Y = R sin Θ.
Question 12. Give an example of two dependent random variables X and Y such that
their linear correlation coefficient ρXY = 0.
2