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AMS 570
Homework 4
Spring 2022
Due March 8
1. Let X amd Y be random variables with joint pdf f (x, y) = e−x−y , x > 0, y > 0, zero elsewhere.
If Z = X + Y , answer the following questions.
(a) Compute P (Z ≤ 0), P (Z ≤ 6), and more generally P (Z ≤ z).
(b) Find the pdf of Z.
2. Let f1|2 (x1 |x2 ) = c1 x1 /x22 , 0 < x1 < x2 < 1, zero elsewhere, and f2 (x2 ) = c2 x42 , 0 < x2 < 1,
zero elsewhere, denote, respectively, the conditional pdf of X1 given X2 = x2 , and the marginal
pdf of X2 . Determine
(a)
(b)
(c)
(d)
The constants c1 and c2 .
The joint pdf of X1 and X2 .
P (1/4 < X1 < 1/2|X2 = 5/8).
P (1/4 < X1 < 1/2).
3. A bivariate population of (X, Y ) is sampled independently on three occasions. On the first, a
random sample of size n0 is taken and only T = min{X, Y } is observed for each pair. On the
second, a random sample of size n1 is taken, and only the X-marginal is observed for each pair.
Finally, a random sample of size n2 is taken, and only the Y -marginal is observed for each pair.
Therefore, the combined set of observations is of the form (T , X, Y ), where T = (T1 , . . . , Tn0 ),
X = (X1 , . . . , Xn1 ) and Y = (Y1 , . . . , Yn2 ). Assume the following two-parameter probability
model for (X, Y):
]
[
1 1/δ
1/δ δ
P (X > x, Y > y) = exp − (x + y ) ,
θ
x > 0, y > 0, θ > 0, 0 < δ ≤ 1 with unknown parameters θ and δ.
(a) Present the joint pdf of (T , X, Y ).
(b) Is the above family an exponential family? Justify your answer.
4. Let X, Y and Z have the joint pdf
( 2
)[
( 2
)]
x + y2 + z2
x + y2 + z2
(2π)−3/2 exp −
1 + xyz exp −
,
2
2
where −∞ < x < ∞, −∞ < y < ∞ and −∞ < z < ∞. Show that X, Y and Z are pairwise
independent and that each pair has a bivariate normal distribution.
5. let X1 and X2 have the joint pdf f (x1 , x2 ) = x1 + x2 , 0 < x1 < 1, 0 < x2 < 1. Find the
conditional mean and variance of X2 given X1 = x1 , 0 < x1 < 1.
6. Let X ∼ Poisson(100). Use Chebyshev’s inequality to determine a lower bound for
P (75 < X < 125).
7. Casella and Berger Exercise 3.29
8. Casella and Berger Exercise 4.10
9. Casella and Berger Exercise 4.16