Random Variables – Examples (with solutions)
1. Let
X
be the number of televisions in an apartment, to be randomly selected in a small
town. Suppose that
X
has probability mass function:
x
0
1
2
p
(
x
)
0.2
0.7
0.1
(a) Compute the mean/expected value of
X
.
Solution:
Let
μ
X
be the mean of
X
, using the formula for discrete random
variables:
μ
X
=
E
(
X
) =
X
x
xp
(
x
)
= 0
*
p
(0) + 1
*
p
(1) + 2
*
p
(2)
= 0
*
(0
.
2) + 1
*
(0
.
7) + 2
*
(0
.
1)
= 0
.
9
(b) What is the probability that this apartment will have at most 2 televisions?
Solution:
P
(
X
≤
2) =
X
x
≤
2
p
(
x
)
=
p
(0) +
p
(1) +
p
(2)
= 0
.
2 + 0
.
7 + 0
.
1 = 1
Notice that it is impossible for realizations of
X
to be greater than 2.
(c) What is the probability that this apartment will have either exactly 0 televisions
or exactly 2 televisions?
Solution:
We are computing the probability of the union of the events:
X
= 0
and
X
= 2, both of which are disjoint, thus:
P
((
X
= 0)
∪
(
X
= 2)) =
P
(
X
=
0) +
P
(
X
= 2). Equivalently, we can write:
P
((
X
= 0)
∪
(
X
= 2)) =
X
(
x
=0)
∪
(
x
=2)
p
(
x
)
=
p
(0) +
p
(2)
= 0
.
2 + 0
.
1
= 0
.
3
1
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2. Let
X
be the amount of time (in minutes) it will take to complete and handin an
exam (where the exam must be handedin no later than 50 minutes from the start).
Suppose that
X
has a probability density function (pdf):
f
(
x
) =
1
20
if 30
≤
x <
50
0
otherwise
(a) Compute the probability this exam will be handedin before 20 minutes have
passed.
Solution:
Since the area under
f
over
x
≤
20 is zero, we have that:
P
(
X
≤
20) = 0
(b) Compute the probability this exam will be handedin when exactly 30 minutes
have passed.
Solution:
The area under
f
over the point
x
= 30, is the area of a line, which
is zero.
P
(
X
= 30) = 0
In general, if
X
is a continuous random variable, the probability that it equals
any constant is always zero.
(c) Compute the probability this exam will be handed in before 32 minutes have
passed.
Solution:
The area under
f
over
x
≤
32 is equal to the area under
f
over
30
≤
x
≤
32, which is a rectangle of width 2 and height 1/20, hence:
P
(
X
≤
32) = 2
*
(1
/
20) = 1
/
10
3. A 6sided die is to be rolled independently 10 times. Let
X
count the number of even
numbers we will roll.
(a) What is the name of
X
’s distribution? (specify parameter(s)).
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 Spring '08
 Staff
 Normal Distribution, Probability, Probability theory, probability density function, Jim, Cumulative distribution function

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