LAB PROBLEMS
1. Suppose that your friend has a strange habit when
eating M&M’s. He will continue eating M&M’s un
immediately stop. Let
X
your friend consumes in one sitting. If we assume
PMF of
X
is
p
X
(
x
) =
±
x

1
x

3
²
0
.
20
3
0
.
80
x

3
where
x
= 3
,
4
,
5
,...
. What is the probability that
snack?
answer: 0.0500278
2. Let
W
be a random random variable with PMF
given by:
p
W
(
w
) =
.
7
if
w
=

2
.
15 if
w
= 0
.
05 if
w
= 1
.
1
if
w
= 5
Find
Pr
(
W
= 0
∪
W
= 1),
Pr
(
W
= 0
∩
W
= 1)
and
Pr
(
W <
1

W
≥
0) .
answer: .2; 0; 1/2
3. FIG. 1 shows the PMF of a random variable
X
.
Find
P
(
X > .
7)
answer: .3
0.4
0.6
0.8
1
x
p(x)
FIG. 1. The above plot shows the PMF of a random variable
X
.
4. Consider a box that contains a bunch of tickets:
each ticket has a 1 or 0 written on it. Let
p
denote
the proportion of tickets with a 1 written on it.
Now randomly pick a ticket from this box and let
the number be denoted
Y
. Find
P
(
Y
= 1),
EY
and
SD
(
Y
).
answer:
p
;
p
;
p
p
(1

p
)
5. Suppose
X
has the following PMF:
p
X
(
x
) =
(
.
7 if
x
=

2
.
3 if
x
= 0
Find the variance of
X
using the formula
∑
x
(
x

EX
)
2
p
X
(
x
) and the formula
E
(
X
2
)

(
EX
)
2
and comment which one is easier
to use.
answer: .84
6. If
EX
= 1 and var(
X
) = 2
.
5 ﬁnd
EY
where
Y
=
(2

X
)
2
.
answer: 3.5
PRACTICE PROBLEMS
1. In the dice game Yahtzee, one objective is to roll
ﬁve sixsided dice simultaneously and try to get
them to all show the same number. Suppose that
your friends allow you to continue rolling all ﬁve
dice until all ﬁve dice match. Let
X
be the number
of turns it takes you to roll ﬁve of the same number.
The PMF of
X
is given by
p
X
(
x
) =
±
6
4

1
6
4
²
x

1
1
6
4
where
x
= 1
,
2
,
3
,...
. What is the probability that
you can stop immediately after your 3rd roll?
answer:
(6
4

1)
2
(6
4
)
3
= 0.00077
2. The number of times a certain web server is ac
cessed per minute (
X
) can be modeled by the fol
lowing PMF.
p
X
(
x
) =
2
.
5
x
e

2
.
5
x
!
where
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Drake
 Statistics, Probability mass function

Click to edit the document details