chapter 2: random variables -...
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Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 1
Chapter 2: Random Variables
Section 2.1: Discrete Random Variables
Problem (01): (a) Verify that the following functions are probability mass functions.
(b) Determine the requested probabilities:
𝑥 -2 -1 0 1 2
𝑓(𝑋 = 𝑥) 1/8 2/8 2/8 2/8 1/8
(1) P(X ≤ 2)
(2) P(X > -2)
(3) P(-1 ≤ X ≤ 1)
(4) P(X ≤ -1 or X = 2)
(Problem 1.1.1 in textbook)
Solution:
∑ 𝑓(𝑋 = 𝑥) =1
8+
2
8+
2
8+
2
8+
1
8= 1.00
≤
≤ ≤
≤
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 2
Problem (02): For the following function, determine:
𝑓(𝑥) =2𝑥 + 1
25 𝑤ℎ𝑒𝑟𝑒 𝑥 = 0, 1, 2, 3, 4
(1) P(X = 4)
(2) P(X ≤ 1)
(3) P(2 ≤ X < 4)
(4) P(X > -10)
(Problem 1.1.1 in textbook)
Solution:
𝑥
𝑓(𝑋 = 𝑥) =2𝑥 + 1
25
≤
≤
Problem (03): An office has four copying machines, and the random variable X measures
how many of them are in use at a particular moment in time. Suppose that:
P(X = 0) = 0.08, P(X = 1) = 0.11, P(X = 2) = 0.27, and P(X = 3) = 0.33.
(a) What is P(X = 4)?
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 3
(b) Draw a line graph of the probability mass function.
(c) Construct and plot the cumulative distribution function.
(Problem 2.1.1 in textbook)
Solution:
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 4
≤
Problem (04): A company has five warehouses, only two of which have a particular
product in stock. A salesperson calls the five warehouses in a random order
until a warehouse with the product is reached. Let the random variable X
be the number of calls made by the salesperson, and calculate its
probability mass function and cumulative distribution function.
(Problem 2.1.9 in textbook)
Solution:
𝒙𝒊
𝒑𝒊2
5
3
5×
2
4=
3
10
3
5×
2
4×
2
3=
1
5
3
5×
2
4×
1
3×
2
2=
1
10
𝑭(𝒙𝒊)2
5
7
10
9
101.0
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 5
Problem (05):
A random variable X has probability mass function of:
𝑓(𝑥) =1
𝐴(6 − 2𝑥) for 𝑥 = 0,1,2
(a) What is the value of A?
(b) Compute and sketch the cumulative distribution function.
(Question 5: in Midterm Exam 2009)
Solution:
∫ 𝑓(𝑥)2
0
= 1.0
∫1
𝐴(6 − 2𝑥)
2
0
= 1.0
∑1
𝐴(6 − 2𝑥)
1
0
= 1.0
[1
𝐴(6 − 2 × 0) +
1
𝐴(6 − 2 × 1) +
1
𝐴(6 − 2 × 2)] = 1.0
≤
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 6
Problem (06): A random variable (X) has a probability mass function of:
𝑓(𝑥) =8
7(
1
2)
𝑥 𝑓𝑜𝑟 𝑥 = 1, 2, 3
(a) (1 point) Verify that this is a valid probability mass function.
(b) (4 point) find the following probabilities.
(1) P(X ≤ 1)
(2) P(X > 2)
(3) P(1 < X < 6)
(4) P(X ≤ 1 or X > 1)
(Question 1: (5 points) in Midterm Exam 2011)
Solution:
∫ 𝑓(𝑥)3
1
= 1.0 ( 𝒄𝒉𝒆𝒄𝒌 ? ! )
∫ 𝑓(𝑥)3
1
= ∫8
7(
1
2)
𝑥3
1
= ∑8
7(
1
2)
𝑥3
1
= [8
7(
1
2)
1
+8
7(
1
2)
2
+8
7(
1
2)
3
]
= 1.0 (𝑂𝑘)
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 7
𝒙
𝒇(𝒙) =𝟖
𝟕(
𝟏
𝟐)
𝒙
≤
≤