Transcript Probability Distributions

Chapter 6
Discrete
Probability
Distributions
DEFINITIONS

Random variable
a variable (typically represented by x) that has a
single numerical value, determined by chance, for
each outcome of a procedure

Discrete random variable
either a finite number of values or countable number of values,
where “countable” refers to the fact that there might be infinitely
many values, but they result from a counting process

Continuous random variable
infinitely many values, and those values can be associated with
measurements on a continuous scale in such a way that there
are no gaps or interruptions
EXAMPLE
Distinguishing Between Discrete and
Continuous Random Variables
Determine whether the following random variables are
discrete or continuous. State possible values for the
random variable.
(a) The number of light bulbs that burn out in a room of
10 light bulbs in the next year.
Discrete; x = 0, 1, 2, …, 10
(b) The number of leaves on a randomly selected Oak
tree.
Discrete; x = 0, 1, 2, …
(c) The length of time between calls to 911.
Continuous; t > 0
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DEFINITIONS
 Probability distribution
a description that gives the probability for each value of
the random variable; often expressed in the format of a
graph, table, or formula
EXAMPLE A Discrete Probability Distribution
The table to the right
shows the probability
distribution for the
random variable X,
where X represents the
number of DVDs a
person rents from a
video store during a
single visit.
x
0
P(x)
1
2
3
4
0.58
0.22
0.10
0.03
5
0.01
0.06
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RULES FOR
PROBABILITY DISTRIBUTION
 P(x) = 1
where x assumes all possible values.
0  P(x)  1
for every individual value of x.
EXAMPLE Identifying Probability Distributions
Is the following a probability distribution?
x
0
1
2
P(x)
3
4
5
0.10
0.30
0.04
0.16
0.18
0.22
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A probability histogram is a
histogram in which the horizontal axis
corresponds to the value of the random
variable and the vertical axis represents
the probability of that value of the
random variable.
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Graphs
The probability histogram is very similar to a relative frequency histogram,
but the vertical scale shows probabilities.
EXAMPLE Drawing a Probability Histogram
Draw a probability histogram of
the probability distribution to the
right, which represents the
number of DVDs a person rents
from a video store during a single
visit.
Probability
DVDs Rented at a Video Store
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Number of DVDs Rented
4
5
x
P(x)
0
0.06
1
0.58
2
0.22
3
0.10
4
0.03
5
0.01
Mean, Variance and
Standard Deviation of a
Probability Distribution
µ =  [x • P(x)]
Mean
 =  [(x – µ) • P(x)]
Variance
2
2
=
 [(x – µ) • P(x)]
2
Standard Deviation
EXAMPLE Computing the Mean of a Discrete
Random
Variable
Compute the mean of the probability
distribution to the right, which
represents the number of DVDs a person
rents from a video store during a single
visit.
x
P(x)
0
0.06
1
0.58
2
0.22
3
0.10
4
0.03
5
0.01
 X   x  P( x)
 0(0.06)  1(0.58)  2(0.22)  3(0.10)  4(0.03)  5(0.01)
 1.49
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DEFINITION
Because the mean of a random variable represents what
we would expect to happen in the long run, it is also
called the expected value, E(X), of the random variable.
The expected value of a discrete random
variable is denoted by E, and it represents the
average value of the outcomes. It is obtained
by finding the value of  [x • P(x)].
E =  [x • P(x)]
EXAMPLE Computing the Expected Value of a
Discrete Random Variable
A term life insurance policy will pay a beneficiary a certain sum of
money upon the death of the policy holder. These policies have
premiums that must be paid annually. Suppose a life insurance
company sells a $250,000 one year term life insurance policy to a 49year-old female for $530. According to the National Vital Statistics
Report, Vol. 47, No. 28, the probability the female will survive the
year is 0.99791. Compute the expected value of this policy to the
insurance company.
x
P(x)
Survives
Does not
survive
$530
0.99791
$530 – $250,000 =
-$249,470
1- 0.99791 =
0.00209
E(X) = 530(0.99791) + (-249,470)(0.00209)
= $7.50
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EXAMPLE Computing the Variance and Standard
Deviation
of a Discrete Random Variable
Compute the variance and standard
deviation of the following probability
distribution which represents the
number of DVDs a person rents from
a video store during a single visit.
2
x


x


P( x)




x  X
X
X
x
P(x)
0
0.06
1
0.58
2
0.22
3
0.10
4
0.03
5
0.01
2
x
0
1
2
3
4
5
P(x)
0.06
0.58
0.22
0.1
0.03
0.01
-1.43
-0.91
-1.27
-1.39
-1.46
-1.48
2.0449
0.8281
1.6129
1.9321
2.1316
2.1904
0.122694
0.480298
0.354838
0.19321
0.063948
0.021904
 X2    x   X  P( x)
2
 1.236892
 X  1.236892
 1.11
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Next:
Section 6.2 The Binomial Probability Distribution
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