Section 7.1
Random Variables and Probability Distributions
A rule that assigns a number to each outcome of an experiment is called a random
variable. Capital letters are often used to represent random variables.
For example, a random variable X can represent the sum of the face values of two sixsided dice. The random variable may take on any number in the set {2, 3, …, 12}.
We can construct the probability distribution associated with a random variable.
If x 1 , x 2 , x 3 ,…, x n are values assumed by the random variable X with associated
probabilities P(X= x 1 ) = p 1 , P(X= x 2 ) = p 2 , …, P(X= x n ) = p n , respectively, then the
probability distribution of X may be expressed in the following way.
x
P(X = x)
x1
x2
.
.
.
xn
p1
p2
.
.
.
pn
We can also graphically represent the probability distribution of a random variable.
A bar graph which represents the probability distribution of a random variable is called a
histogram.
Example 1: Given the following histogram, calculate the probability that x = 3.
Section 7.1 – Random Variables and Probability Distributions
1
Example 2: The rates paid by 25 financial institutions on a certain day for money-market
deposit accounts are shown in the accompanying table:
Rate, %
2.95
3.00
3.15
3.25
Number of
Institutions
3
7
7
8
a. Let the random variable X denote the interest paid by a randomly chosen financial
institution on its money-market deposit accounts and find the probability distribution
associated with these data.
b. Draw the histogram associated with these data.
c. Find:
P(X > 3.00)
P(3.00 < X < 3.25)
Section 7.1 – Random Variables and Probability Distributions
2
M 1313 Popper number 19
Use for questions 1, 2 and 3.
An automobile manufacturer obtains the microprocessors used to regulate fuel
consumption in its automobiles from three microelectronic firms: A, B, and C. The
quality-control department of the company has determined that 1% of the
microprocessors produced by firm A are defective, 2% of those produced by firm B are
defective, and 1.5% of those produced by firm C are defective. Firms A, B, and C supply
45%, 25%, and 30%, respectively, of those microprocessors used by the company. An
automobile is selected at random. Draw a tree diagram.
1. What is the probability it was manufactured at firm B and it was found
to be defective?\
A. .0050 B. .0150 C. .0140
D. .0200
2. What is the probability it is defective?
A. .0038
B. .0150
C. .0140
D. .0200
3. What is the probability it was defective given it was manufactured at firm C?
A. .0038
B. .0150
C. .0140
D. .0200
Use for questions 4 and 5.
An urn contains 10 red and 13 blue marbles. Two marbles are chosen at
random, in succession and without replacement. Draw tree diagram.
4. What is the probability a marble is blue?
A. .4348
B. .5652
C. .5504
D. .5455
5. What is the probability that the first marble was red, given that the second one was
blue?
A. .4546
B. .5652
C. .5504
D. .5455 Math 1313 Popper Number 17
1. A new tax business, ABC taxes, will purchase a copying machine. After
speaking with their financial advisor, they find that the copying machine will
cost them $4,300 in 2 years. The account they will invest in earns 5% per
year compounded quarterly. In order to pay cash for the machine, how much
should they deposit quarterly in this account for 2 years? What kind of
problem is it?
A. Amortization
C. Sinking Fund
B. Future Value of an annuity
D. Present Value of an annuity
2. A library decides to buy a new computer system through Amex Company.
They make a down payment of $4,000. If Amex Company charges 5 % per
year compounded quarterly for 2 years, and the library's quarterly payments
are $10,000, what is the purchase price of the computer system?
A. $ 79,681.24 B. $ 71,681.24 C. $ 83,588.88 D. $ 75,681.24
.
Use for questions 3 and 4
U
A
3
B
13
2
1
9
5
C
11
7
3. Find n(B I (A U C)c ) .
A. 9
B. 2
C. 12
D. 14
4. Find n(A ∪ B c )
A. 13
B. 27
C. 44 D. 40
5. This test is all multiple choice so the score you see in CASA is
your grade for this test.
A. yes
M1313
Assignment 18
Bubble PS#
1.Let S be a sample space and E and F two events of S. Given that P(E) =
0.46, P(F ) = 0.38 and P(E ∩ F) = .04 . Find P(E |F).
A. .1053
2. Let S be a sample space and E and F two events of S. Given that
P(E) = 0.42, P(F ) = 0.31 and P(F|E) = 0.3333. Find P(E I F).
A. .1400
3. Let S be a sample space and E and F are two independent events
of S. If P(E) = .32 and the P(F) = .21. Find the P(E |F).
A. .3200
Use the following to answer question 4 and 5. A tree diagram of drawing
two marbles that are blue and white without replacement in succession.
6/10 B2
7/11
Start
B1
4/10
W2
7/10 B2
4/11
W1
3/10
W2
4. Find the probability that both marbles drawn are blue.
A. .3818
5. Find the probability the second marble drawn was blue given that
the first marble was white.
A. .7000