STATISTICS 244
Problem Set 3
9:0010:20 TTh
Due Thursday January 26
1.
Assume the random variable X has the Bernoulli distribution (i.e., binomial with
n = 1):
P(X = 1) =
q
= 1 – P(X = 0).
(a) Find E(
).
(b) Find E(X
3
). (c) Find E(X
10
).
(d) Find Var(X
3
). (e) Find E(3X
3
+4).
(f) Find Var(3X
3
+4).
2.
Consider the following
bivariate distribution:
Y
2
3
4
5
1
.1
.1
.0
.0
X
2
.0
.2
.2
.1
3
.0
.0
.1
.2
(a) Find the marginal distributions of X and Y.
(b) Find the conditional distribution of Y
given X = 1. Find the conditional distribution of Y given X = 2.
(c) Find E(Y).
(d) Find the covariance of X and Y.
3.
Suppose an urn contains 3 tickets numbered “1”, 3 tickets numbered “2”, 2 ticket
numbered “3”, and 1 ticket numbered “4”.
A student draws a ticket at random and notes the
number, X.
The student then returns the ticket to the urn, shakes it up, and draws again,
noting the number, Y.
Let Z = the minimum of X and Y.
(a) Find the probability distribution of X, the cumulative distribution of X, and graph both.
(b) Find the probability distribution of Z. (c) Find E(X), E(Y), E(Z).
(d) Find the bivariate
probability function p(x, z) of X and Z, and the covariance of X and Z.
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 Fall '08
 DRTON
 Statistics, Bernoulli, Binomial

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