STATS401HW7_Sol.pdf - HW7 Sol Stats 401 Instruction Team...

HW7 Sol
Stats 401 Instruction Team
February 16, 2019
Question 1
Let U and V be iid (independent and identically distributed) with mean μ and standard deviation σ .
a. What is V ar (
1
4
U +
3
4
V ) ?
b. What is V ar (
1
2
U +
2
V ) ?
c. Which has bigger variance?
Question 2
Consider the distribution with pmf given by: P ( X = - 1) = 0 . 4 and P ( X = 1) = 0 . 6 . Let X
1
and X
2
be
independent and identically distributed random variables with this distribution.
a. What is E ( X
1
) and V ar ( X
2
) ?
b. Let S = X
1
+ X
2
, the sum of X
1
and X
2
. What is the pmf of S ? (Determine first the possible values
of S and then obtain the corresponding probabilities. This is the sampling distribution of
c. Let Y =
S
2
=
2
. What is the distribution (that is, the pmf) of Y ? (This is the sampling distribution
of the sample mean based on two observations for this case.)
1

d. Compute E(Y) and Var(Y) and compare them to your answer in part (a). What relationship do you
observe?
Question 3
Recall the game in Question 3, Homework 6 (repeated briefly here: Joe Smith plays a betting game with his
friends. He puts in one dollar to join the game. If he wins the game, he gets 2. If he loses, he gets nothing,
so his gain is -1). Joe Smith is planning to play the game 100 times. The 100 outcomes will be independent
of each other.
a. What is his expected total gain if he plays 100 times?