UW-Madison 431 - Introduction to Probability Theory Spring 2009
Second Midterm Solutions
1. (12 points) In each of the following cases identify the name of the distribution of the random
variable Xand determine the probability mass function.
(a) We have an urn with 5 red and 18 blue balls and we pick 4 balls without replacement. We
denote the number of red balls in the sample by X.
This is a hypergeometric random variable with parameters n= 4, N = 23 and m= 5. the
probability mass function is
P(X=k) = ¡5
k¢¡18
5−k¢
¡23
4¢,0≤k≤4
(b) We have an urn with 5 red and 18 blue balls and we pick 4 balls with replacement. We
denote the number of red balls in the sample by X.
This is a binomial random variable with parameters n= 4, p =5
23 . The probability mass
function is
P(X=k) = µ4
k¶µ5
23¶kµ18
23¶4−k
,0≤k≤4
(c) We roll a fair die until we get a three or a four. Xdenotes the number of rolls needed.
This is a geometric random variable with parameter p=2
3. The probability mass function
is
P(X=k) = 2
3·1
3k−1k≥1
2. (18 points) The discrete random variable Xcan take on the possible values −1,0,1,2 and it
has a probability mass function p(a) = 1
10 (a2+ 1).
(a) What is the value of the cumulative distribution function of Xat 0.3?
P(X≤0.3) = P(X=−1) + P(X= 0) = 1
5+1
10 =3
10
(b) Compute the expectation of X.
E[X] =
2
X
i=−1
p(a)a= (−1)1
5+ 0 ·1
10 + 1 ·1
5+ 2 ·1
2= 1
(c) Compute the variance of X.
We first compute the second moment:
E[X2] =
2
X
i=−1
a2p(a) = 1 ·1
5+ 0 ·1
10 + 1 ·1
5+ 4 ·1
2=12
5
which gives
V ar[X] = E[X2]−E[X]2=7
5
1
