Transcript Chapter 7: Random Variables and Probability Distributions

Chapter 7: Random Variables
and Probability Distributions
Section 7.1: Random
Variables
• Random Variable – a numerical variable whose
value depends on the outcome of a chance
experiment.
• A random variable associates a numerical value
with each outcome of a chance experiment.
• Discrete – A random variable is discrete if its set
of possible values is a collection of isolated
points on the number line.
• Continuous – The variable is continuous if its set
of possible values includes an entire interval on
the number line.
Example
• Consider an experiment in which the type
of car, new (N) or used (U), chosen by
each of three successive customers at a
discount car dealership is noted. Define a
random variable x by
x = number of customers purchasing a new car
The eight possible outcomes are listed
below:
Outcome: UUU NUU UNU UUN NNU NUN UNN NNN
x value:
0
1
1
1
2
2
2
3
There are only four possible x values – 0, 1, 2, and 3 – and
these are isolated points on the number line. Thus, x is a
discrete random variable.
Example
• In an engineering stress test, pressure is applied
to a thin 1-ft long bar until the bar snaps. The
precise location where the bar will snap is
uncertain. Let x be the distance from the left end
of the bar to the break. Then x = 0.25 is on
possibility, x = 0.9 is another, and in face any
number between 0 and 1 is a possible value of
x. This set of possible values is an entire interval
on the number line, so x is a continuous random
variable.
Section 7.2: Probability
Distributions for Discrete
Random Variables
• Probability distribution of a discrete random
variable x – gives the probability associated with
each possible x value. Each probability is the
limiting relative frequency of occurrence of the
corresponding x value when the chance
experiment is repeatedly performed.
• Common ways to display a probability
distribution for a discrete random variable are a
table, a probability histogram, or a formula.
Example
• Suppose that each of four randomly selected customers
purchasing a hot tub at a certain store chooses either an
electric (E) or a gas (G) model. Assume that these
customers make their choices independently of one
another and that 40% of all customers select an electric
model. This implies that for any particular one of the four
customers, P(E) = .4 and P(G) = .6. One possible
outcome is EGGE, where the first and fourth customers
select electric models and the other two choose gas
models. Because the customers make their choices
independently, the multiplication rule for independent
events implies…
P(EGGE) = P(1st chooses E and 2nd
chooses G and 3rd chooses G and 4th
chooses E)
= P(E)P(G)P(G)P(E)
= (.4)(.6)(.6)(.4)
= .0576
Let x = the number of electric hot tubs
purchased by the four customer
Outcomes and Probabilities
Outcome
Probability x value
Outcome
Probability x value
GGGG
EGGG
GEGG
GGEG
GGGE
EEGG
EGEG
EGGE
.1296
.0864
.0864
.0864
.0864
.0576
.0576
.0576
GEEG
GEGE
GGEE
GEEE
EGEE
EEGE
EEEG
EEEE
.0576
.0576
.0576
.0384
.0384
.0384
.0384
.0256
0
1
1
1
1
2
2
2
2
2
2
3
3
3
3
4
• The probability distribution of x is easily
obtained from this information. Consider
the smallest possible x value, 0. The only
outcome for which x = 0 is GGGG, so
p(0) = P(x = 0) = P(GGGG) = .1296
• Let’s find the 4 other outcomes:
p(1) = P(x = 1) = P(EGGG or GEGG or GGEG or GGGE)
= P(EGGG) + P(GEGG) + P(GGEG) + P(GGGE)
= .0864 + .0864 + .0864 + .0864
= 4(.0864)
= .3456
p(2) = 6(.0576) = .3456
p(3) = 4(.0384) = .1536
p(4) = .0256
• Properties of Discrete Probability Distributions:
1. For every possible x value, 0 ≤ p(x) ≤ 1.
2.
 p ( x)  1
all x values
Example
• A consumer organization that evaluates
new automobiles customarily reports the
number of major defects on each car
examined. Let x denote the number of
major defects on a randomly selected car
of a certain type. A large number of
automobiles were evaluated, and a
probability distribution consistent with
theses observations is:
x
0
1
2
3
4
5
6
7
8
9
10
p(x) .041 .130 .209 .223 .178 .114 .061 .028 .011 .004 .001