Chapter Six Discrete Probability Distributions

1
Chapter SixDiscrete Probability Distributions

• 6.1
• Probability Distributions

2
Objectives

• Distinguish between discrete and continuous
  random variables.
• Construct identify probability distributions.
• Construct probability histograms
• Compute and interpret the mean, variance,
  standard deviation and expected value of a
  discrete random variable.

3
Random Variables
When experiments are conducted in such a way that
the outcome is a numerical result, we say the
outcome is a random variable. A random variable
is a numerical measure of the outcome from a
probability experiment, so its value is
determined by chance. Random variables are
denoted using letters such as X.
4
Random Variables
A discrete random variable is a random variable
that has values that result from counting. A
continuous random variable is a random variable
that has values that result from measurement.
5
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. (b) The number of leaves on a randomly
selected Oak tree. (c) The length of time between
calls to 911.
6
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 (b) The number of leaves on a randomly
selected Oak tree. (c) The length of time between
calls to 911.
7
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 (b) The number of leaves on a randomly
selected Oak tree. Discrete (c) The length of
time between calls to 911.
8
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 (b) The number of leaves on a randomly
selected Oak tree. Discrete (c) The length of
time between calls to 911. Continuous
9
Probability Distribution

• A probability distribution provides the possible
  values of the random variable and their
  corresponding probabilities.
• A probability distribution can be in the form of
  a table, graph or mathematical formula.

10
Probability Distribution
The table below 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.
11
This means that the probabilities of all the
possible outcomes must equal to 1.
This means that the probabilities of each outcome
must be between 0 and 1, inclusive.
12
EXAMPLE Identifying Probability
Distributions Is the following a probability
distribution?
13
EXAMPLE Identifying Probability
Distributions Is the following a probability
distribution? No, because the probabilities
DO NOT add up to 1.
14
EXAMPLE Identifying Probability
Distributions Is the following a probability
distribution?
-
15
EXAMPLE Identifying Probability
Distributions Is the following a probability
distribution? No, because the probabilities
are NOT between 0 and 1, inclusive.
-
16
EXAMPLE Identifying Probability
Distributions Is the following a probability
distribution?
17
EXAMPLE Identifying Probability
Distributions Is the following a probability
distribution? Yes, because the probabilities
add up to 1 AND each probability is between 0 and
1, inclusive.
18
Probability Histogram
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.
19
EXAMPLE Drawing a Probability
Histogram Draw a probability histogram of the
following probability distribution which
represents the number of DVDs a person rents from
a video store during a single visit.
20
EXAMPLE Drawing a Probability Histogram
21
Mean of a Discrete Random Variable
22
EXAMPLE The Mean of a Discrete Random
Variable Compute the mean of the following
probability distribution which represents the
number of DVDs a person rents from a video store
during a single visit.
23
EXAMPLE The Mean of a Discrete Random Variable
The Mean is the SUM of the last column. So ux
1.49
24
(No Transcript)
25
EXAMPLE The Mean of a Discrete Random Variable
The following data represent the number of DVDs
rented by 100 randomly selected customers in a
single visit. Compute the mean number of DVDs
rented.
26
EXAMPLE The Mean of a Discrete Random Variable
27
(No Transcript)
28
Expected Value
NOTE This is the same as finding the mean!
29

• EXAMPLE Expected Value
• 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.

30

• EXAMPLE Expected Value
• Suppose a life insurance company sells a 250,000
  one year term life insurance policy to a
  49-year-old female for 520.
• Assume the probability the female will survive
  the year is 0.99791.
• Compute the expected value of this policy to the
  insurance company.

31

• EXAMPLE Expected Value
• The only 2 outcomes are survival or death.
• Probability of survival is .99791
• So the probability of death is 1- .99791 0.00209
• The Expected Value is

32

• EXAMPLE Expected Value
• The only 2 outcomes are survival or death.
• Probability of survival is .99791
• So the probability of death is 1- .99791 0.00209
• The Expected Value is
• E(X) (250,000 .00209) (0 .99791)
• E(X) 522.50

33
Variance Standard Deviation of a Discrete
Variable
34
EXAMPLE Variance and Standard
Deviation 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.
35
EXAMPLE Variance and Standard
Deviation Using your calculator we find that
for The variance of the discrete random
variable is sx2 0.8989 The std. deviation of
the discrete random variable sx 0.9327