an appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. let x = the amount of storage space purchased by the next customer to buy a freezer. suppose that x has pmf x 13.5 15.9 19.1 p(x) 0.2 0.5 0.3
Answers
Answer:
An appliance dealer sells three different models of upright freezers having 11.5, 14.9, and 19.1 cubic feet of storage space. Let x = the amount of storage space purchased by the next customer to buy a freezer. Suppose that x has the following probability distribution.
x p(x)
11.5 .2
14.9 .5
19.1 .3
(a) Calculate the mean and standard deviation of x. (Enter your answers to three decimal places.)
(b) If the price of the freezer depends on the size of the storage space, x, such that Price = 25x - 8.5, what is the mean value of the variable Price paid by the next customer?
(c) What is the standard deviation of the price paid? (Enter your answer to the nearest cent.)
Expected Value and Variance of a Random Variable:
The expected value and the standard deviation of a random variable given the distribution or the specific values of the random variable can be found using the general definitions of the expected value as well as the variance. Now, for any equation that is a linear combination of the random variable under consideration, it is possible to calculate the expected value and the variance using the specific properties of both of the factors.
Answer and Explanation: 1
The following information can be found from the given table:
x p(x) x*p(x) x*x*P(x)
11.5 0.2 2.3 26.45
14.9 0.5 7.45 111.005
19.1 0.3 5.73 109.443
SUM: 15.48 246.8980
(a.)
The mean of
x
is given by:
Mean
=
E
[
x
]
=
3
∑
i
=
1
x
×
p
(
x
)
=
15.480
Calculation of the standard deviation of
x
is shown below.
E
[
x
2
]
=
3
∑
i
=
1
x
2
×
p
(
x
)
=
246.8980
The variance of
x
is given by:
V
a
r
[
x
]
=
E
[
x
2
]
−
(
E
[
x
]
)
2
So, the standard deviation of
x
will be:
S
D
[
x
]
=
√
V
a
r
[
x
]
=
√
E
[
x
2
]
−
(
E
[
x
]
)
2
=
√
246.8980
−
15.480
2
=
2.696
(b.)
Let's denote the Price by
P
.
Points to note before we calculate the expected value:
Expected Value of a constant is a constant itself.
Expected Value of the product of a constant and the random variable is the expected value of the random variable multiplied by the constant, that is,
E
[
25
X
]
=
25
×
E
[
X
]
P
=
25
X
−
8.5
E
[
P
]
=
E
[
25
x
−
8.5
]
=
25
E
[
X
]
−
8.5
=
25
×
15.480
−
8.5
=
378.5
(c.)
Points to note before we calculate the variance:
Variance of a constant is
0
.
Variance of the product of a constant and the random variable is the variance of the random variable multiplied by the square of the constant, that is,
E
[
25
X
]
=
25
2
×
V
a
r
[
X
]
P
=
25
X
−
8.5
V
a
r
[
P
]
=
V
a
r
[
25
x
−
8.5
]
=
25
2
V
a
r
[
X
]
−
0
=
625
×
2.696
2
=
4542.76
Therefore, the standard deviation of the price paid is:
S
D
[
P
]
=
√
V
a
r
[
P
]
=
√
4542.76
=
67.4
The standard deviation of the price paid to the nearest cent is
67
.