ENGR 323
3-16
BEAUTIFUL HOMEWORK #4
1 /3
Marketing estimate that a new instrument for the analysis of soils samples will be very
successful, moderately successful, or unsuccessful, with probabilities 0.3, 0.6, and 0.1,
respectively. The yearly revenues associated with a very successful. moderately successful,
or unsuccessful product is $10 million, $5 million, and $1 million, respectively. Let the
random variable X denote the yearly revenue of the product. Determine the probability
mass function of X.
Defining the random variable:
X = the yearly revenue (millions of dollars) of the product
Since the random variable, X, has a finite (or countably infinite) range, it’s a
discrete random variable and the range is X = {1, 5, 10}.
Definition of the probability mass function (Pg. 105) states that the function ƒX(x) is
the probability of x, P(X = x), from the set of possible values of the discrete random
variable X to the interval [0, 1]. For a random variable X, ƒX(x) must satisfied the
following properties:
(1)
ƒX(x) = P(X = x)
(2)
ƒX(x) ≥ 0, ∀ x
(3)
∑ f (x) = 1
X
x
So, properties must be checked and satisfied before the exercise can be
proceed further;
ƒX(1) = P(X = 1) = 0.1
ƒX(5) = P(X = 5) = 0.6
ƒX(10) = P(X = 10) = 0.3
Clearly shown from above functions that for every x ∈ X, ƒX(x) ≥ 0. So, property (1)
and (2) are satisfied.
ƒX(1) + ƒX(5) + ƒX(10) = P(X = 1) + P(X = 5) + P(X = 10)
= 0.1 + 0.6 + 0.1
=1
Property (3) is also satisfied.
Therefore, Probability Mass Function (PMF) is well defined for the exercise 3-16.
Table 1: Summary of PMF of the Discrete Random Variable X.
x
1
5
fX(x)
0.1
0.6
10
0.3
A Probability Mass Function (PMF) graph is generated from Table 1 in Figure 1
below.
ENGR 323
BEAUTIFUL HOMEWORK #4
2 /3
1
0.9
Probability of ƒ(x)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
x (Millions of Dollars)
Figure 1:
Probability mass function (PMF) for Exercise 3-16.
Definition of Cumulative Distribution Function (CDF) of a random discrete variable
X (Pg. 109), denoted as FX(x), is
FX(x) = P(X ≤ x) =
∑ f (x )
xi ≤ x
i
For a discrete random variable X, FX(x) satisfied the following properties.
(1)
FX(x) = P(X ≤ x) =
∑ f (x )
xi ≤ x
i
(2)
0 ≤ FX(x) ≤ 1
(3)
If x ≤ y, then FX(x) ≤ FX(y)
Clearly, properties (1), (2), and (3) are satisfied, so the CDF for Exercise 3-16 is well
defined. The CDF is summarized in the following table:
Table 2: Summary of CDF of X for Exercise 3-16.
x
x<1
5 > x ≥1
10 > x ≥ 5
F(x)
0
0.1
0.7
x > 10
1.0
The Cumulative Distribution Function (CDF) can also be summarized in the
notation below:



FX (x) = 


0
0.1
0.7
1
x< 1
1≤ x <5
5 ≤ x < 10
10 ≤ x
ENGR 323
BEAUTIFUL HOMEWORK #4
3 /3
A Cumulative Distribution Function (CDF) graph is generated from Table 2 in
Figure 2 below:
1.2
Cumulative Distribution Function of F(x)
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-1
0
1
2
3
4
5
6
7
8
9
x (Millions of Dallars)
Figure 2: Cumulative Distribution Function (CDF) for Exercise 3-16.
10
11
12