Download STATISTICS 251 HOMEWORK ASSIGNMENT 8 DUE MONDAY

STATISTICS 251
HOMEWORK ASSIGNMENT 8
DUE MONDAY DECEMBER 1
Problem 1. Let U1 , U2 be independent random variables both uniformly distributed on [0, 1], and
set
M = max(U1 , U2 ) and
N = min(U1 , U2 ).
(a) Find the conditional joint density of (U1 , U2 ) given M ≤ 1/2.
(b) Find the conditional density of N given M = a for any value of a ∈ (0, 1).
(c) Find cov(M, N ).
Problem 2. Let X, Y, Z be independent random variables where X, Y are uniformly distributed
on [0, 1] and Z is an exponential with mean 1.
(a) What is the conditional density of X given XY = t for 0 ≤ t ≤ 1?
(b) What is the conditional density of X given X + Z = t for t ≥ 0?
Problem 3. Let X1 , . . . , Xn be i.i.d. standard Gaussian variables, and let Sk =
1, . . . , n. Let m < n be an integer.
Pk
i=1
Xi for k =
(a) Find the conditional distribution of Sn given Sm = s.
(b) Find the conditional distribution of Sm given Sn = t.
Problem 4. Suppose that in a certain large population mothers’ heights X and daughters’ heights
Y are both normally distributed with mean 65 inches and standard deviation 3 inches. Suppose
further that mothers’ and daughters’ heights are jointly normal with correlation 1/2.
(a) Among all daughters whose mothers were approximately 68 inches tall, what fraction are taller
than their mothers?
(b) What fraction of all mother/daughter pairs have average height above 66 inches?
(c) What is the average height of all mothers with above-average heights, that is, what is
E(X | X > 65)?
Problem 5. Let N be a random variable that takes values in the positive integers 1, 2, . . . . Given
that N = n, toss a fair coin n times, and let X be the number of Heads and Y = N − X the number
of Tails. What is the covariance of X and Y if
(a) N has the Poisson distribution with parameter λ > 0?
(b) N has the geometric distribution
P {N = k} = rk−1 (1 − r)?
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