1. Continuous Probability Distributions
Statistics
Continuous Probability Distributions

2. Continuous Probability Distributions
Uniform Probability Distribution
Normal Probability Distribution
Normal Approximation of Binomial Probabilities
Exponential Probability Distribution

3. Continuous Probability Distributionsx
f (x)
Exponential
f (x) x
Uniform x
f (x)
Normal

4. STATISTICS in PRACTICE
Procter & Gamble (P&G) produces
and markets such products as
detergents, disposable diapers,
bar soaps, and paper towels.
The Industrial Chemicals Division
of P&G is a supplier of fatty alcohols derived from natural substances such as coconut oil and from petroleum- based derivatives.

5. STATISTICS in PRACTICE
The division wanted to know the economic
risks and opportunities of expanding its
fatty-alcohol production facilities, so it
called in P&G’s experts in probabilistic
decision and risk analysis to help.

6. Continuous Probability Distributions
A continuous random variable can assume any value in an interval on the real line or in a collection of intervals.
It is not possible to talk about the probability of the random variable assuming a particular value.
Instead, we talk about the probability of the random variable assuming a value within a given interval.

7. Continuous Probability Distributions
The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function f(x) between x1 and x2.
f(x) ≧0

8. Continuous Probability Distributions
f (x) x
Uniform x1 x2 x
f (x)
Normal x1 x2 x1 x2
Exponential x
f (x)

9. Continuous Probability Distributions
The expected value of a continuous random variable x is:
The variance of a continuous random variable x is:

10. Continuous Probability Distributions
If the random variable x has the density function f(x), the probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be

11. Continuous Probability Distributions
Cumulative probability function:
If the random variable x has the density function f(x), the cumulative distribution function for x ≦x2 is:

12. Uniform Probability Distribution
A random variable is uniformly distributed whenever the probability is proportional to the interval’s length.
The uniform probability density function is:
f (x) = 1/(b – a) for a < x < b
= elsewhere
where: a = smallest value the variable can assume
b = largest value the variable can assume

13. Uniform Probability Distribution
Expected Value of x
E(x) = (a + b)/2
Variance of x
Var(x) = (b - a)2/12

14. Uniform Probability Distribution
Example:
Random variable x = the flight time of an
airplane traveling from Chicago to New
York. Suppose the flight time can be any
value in the interval from 120 minutes to 140
minutes.

15. Uniform Probability Distribution
Assume every 1-minute interval being equally likely, x is said to have a uniform
probability distribution and the probability density function is

16. Uniform Probability Distribution
Example: Uniform Probability Density
Function for Flight Time

17. Uniform Probability Distribution
What is the probability that the flight time is between 120 and 130 minutes? That is, what is ?
Area provides Probability of Flight Time Between 120 and 130 Minutes

18. Uniform Probability Distribution
Applying these formulas to the uniform distribution for flight times from Chicago to New York, we obtain
and σ=5.77 minutes.

19. Uniform Probability Distribution
Example: Slater's Buffet
Slater customers are charged
for the amount of salad they
take. Sampling suggests
that the amount of salad
taken is uniformly
distributed between 5 ounces and 15 ounces.

20. Uniform Probability Distribution
Uniform Probability Density Function
f(x) = 1/10 for 5 < x < 15
= elsewhere
where:
x = salad plate filling weight

21. Uniform Probability Distribution
Expected Value of x
Variance of x
E(x) = (a + b)/2
= (5 + 15)/2
= 10
Var(x) = (b - a)2/12
= (15 – 5)2/12 =

22. Uniform Probability Distribution
for Salad Plate Filling Weight
f(x)
1/10 x 5 10 15
Salad Weight (oz.)

23. Uniform Probability Distribution
What is the probability that a customer
will take between 12 and 15 ounces of salad?
f(x)
P(12 < x < 15) = 1/10(3) = .3
1/10 x 12 5 10 15
Salad Weight (oz.)

24. Normal Probability Distribution
The normal probability distribution is the most important distribution for describing a continuous random variable.
It is widely used in statistical inference.

25. Normal Probability Distribution
It has been used in a wide variety of applications:
Heights
of people
Scientific
measurements

26. Normal Probability Distribution
It has been used in a wide variety of applications:
Test
scores
Amounts
of rainfall

27. Normal Probability Distribution
Normal Probability Density Function
where:
 μ = mean,
 σ = standard deviation,
 π =
e =

28. Normal Probability Distribution
Characteristics
The distribution is symmetric; its skewness
measure is zero. x

29. Normal Probability Distribution
Characteristics
The entire family of normal probability
distributions is defined by its mean μ and
its standard deviation σ .
Standard Deviation s x
Mean m

30. Normal Probability Distribution
Characteristics
The highest point on the normal curve is at the
mean, which is also the median and mode. x

31. Normal Probability Distribution
Characteristics
The mean can be any numerical value:
negative, zero, or positive. x
-10 20

32. Normal Probability Distribution
Characteristics
The standard deviation determines the width of
the curve: larger values result in wider, flatter
curves.
s = 15
s = 25 x

33. Normal Probability Distribution
Characteristics
Probabilities for the normal random variable are
given by areas under the curve. The total area
under the curve is 1 (.5 to the left of the mean and
.5 to the right). .5 .5 x

34. Normal Probability Distribution
Characteristics
of values of a normal random variable
are within of its mean.
68.26%
+/- 1 standard deviation
of values of a normal random variable
are within of its mean.
95.44%
+/- 2 standard deviations
of values of a normal random variable
are within of its mean.
99.72%
+/- 3 standard deviations

35. Normal Probability Distribution
Characteristics
99.72%
95.44%
68.26% x m
m – 3s
m – 1s
m + 1s
m + 3s
m – 2s
m + 2s

36. Standard Normal Probability Distribution
A random variable having a normal distribution
with a mean of 0 and a standard deviation of 1 is
said to have a standard normal probability
distribution.

37. Standard Normal Probability Distribution
The letter z is used to designate the standard
normal random variable.
s = 1 z

38. Standard Normal Probability Distribution
Areas, or probabilities, for The Standard Normal Distribution

39. Standard Normal Probability Distribution

40. Standard Normal Probability Distribution
Example: What is the probability that the z value for the standard normal random variable will be between .00 and 1.00?

41. Standard Normal Probability Distribution

42. Standard Normal Probability Distribution
Example: = ?
Table 6.1 to show that the probability of a z value between z = .00 and z = 1.00 is .3413
the normal distribution is symmetric, therefore, =
= = .6826

43. Standard Normal Probability Distribution
Example: = ?
Table 6.1 to show that the probability of a z value between z = .00 and z = 1.00 is .3413
the normal distribution is symmetric, therefore, =
= = .6826

44. Standard Normal Probability Distribution

45. Standard Normal Probability Distribution
Example: = ? =
the normal distribution is symmetric and
= =

46. Standard Normal Probability Distribution

47. Standard Normal Probability Distribution
Example: = ?
Probability of a z value between z = 0.00 and
z = 1.00 is .3413, and Probability of a z value between z = 0.00 and z =1.58 is
Hence, Probability of a z value between z = 1.00 and z = 1.58 is — =

48. Standard Normal Probability Distribution
Example: = ?
Probability of a z value between z = 0.00 and z = 1.00 is .3413, and Probability of a z value between z = 0.00 and z =1.58 is.4429.
Hence, Probability of a z value between z = 1.00 and z = 1.58 is — = .1016

49. Standard Normal Probability Distribution
Example: find a z value such that the probability of obtaining a larger z value is .10.

50. Standard Normal Probability Distribution
An area of approximately (actually .3997) will be between the mean and z * In terms of the question originally asked, the probability is approximately .10 that the z value will be larger than 1.28.
Standard Normal Probability Distribution

51. Standard Normal Probability Distribution
Converting to the Standard Normal Distribution
We can think of z as a measure of the
number of standard deviations x is from .

52. Standard Normal Probability Distribution
Standard Normal Density Function
where:
z = (x – m)/s
 =
e =

53. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Example: Pep Zone
Pep Zone sells auto parts and supplies including a popular multi-grade
motor oil. When the stock of this
oil drops to 20 gallons, a replenishment order is placed.

54. Standard Normal Probability Distribution
Example: Pep Zone
The store manager is concerned that sales are being lost due to stockouts while waiting
for an order. It has been determined
that demand during replenishment
lead-time is normally distributed
with a mean of 15 gallons and a
standard deviation of 6 gallons.
Pep
Zone
5w-20
Motor Oil

55. Standard Normal Probability Distribution
Example: Pep Zone
The manager would like to know the
probability of a stockout, P(x > 20).
Pep
Zone
5w-20
Motor Oil

56. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Solving for the Stockout Probability
Step 1: Convert x to the standard normal distribution.
z = (x - )/
= ( )/6
= .83
Step 2: Find the area under the standard normal
curve to the left of z = .83.
see next slide

57. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Cumulative Probability Table for
the Standard Normal Distribution
P(z < .83)

58. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Solving for the Stockout Probability
Step 3: Compute the area under the standard normal
curve to the right of z = .83.
P(z > .83) = 1 – P(z < .83) = =
Probability
of a stockout
P(x > 20)

59. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Solving for the Stockout Probability
Area = =
Area = .7967 z
.83

60. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Standard Normal Probability Distribution
If the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be?

61. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Solving for the Reorder Point
Area = .9500
Area = .0500 z
z.05

62. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Solving for the Reorder Point
Step 1: Find the z-value that cuts off an area of .05
in the right tail of the standard normal
distribution.
We look up the complement of the tail area ( = .95)

63. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Solving for the Reorder Point
Step 2: Convert z.05 to the corresponding value
of x.
x =  + z.05
 = (6)
= or 25
A reorder point of 25 gallons will place the
probability of a stockout during leadtime at
(slightly less than) .05.

64. Standard Normal Probability Distribution
Pep
Zone
5w-20
Motor Oil
Standard Normal Probability Distribution
Solving for the Reorder Point
By raising the reorder point from 20
gallons to 25 gallons on hand, the probability
of a stockout decreases from about .20 to .05.
This is a significant decrease in the chance
that Pep Zone will be out of stock and unable
to meet a customer’s desire to make a purchase.

65. Normal Approximation of Binomial Probabilities
When the number of trials, n, becomes large,
evaluating the binomial probability function by hand or with a calculator is difficult
The normal probability distribution provides an easy-to-use approximation of binomial probabilities where n > 20, np > 5,
and n(1 - p) > 5.

66. Normal Approximation of Binomial Probabilities
Set
 = np
Add and subtract 0.5 (a continuity correction
factor) because a continuous distribution is
being used to approximate a discrete
distribution. For example,
P(x = 10) is approximated by P(9.5 < x < 10.5).

67. Normal Approximation of Binomial Probabilities
Example: To find the binomial probability of 12 successes in 100 trials and p = .1.
n= 100, p = .1. np = 10 > 5 and
n(1 - p) = (100)(.9) =90 > 5.
Mean, np = (100)(.1) = 10 > 5, Variance, np(1 - p) = (100)(.1)(.9) =9. Compute the area under the corresponding normal curve between 11.5 and 12.5.

68. Normal Approximation of Binomial Probabilities

69. Exponential Probability Distribution
The exponential probability distribution is useful in describing the time it takes to complete a task.
The exponential random variables can be used to describe:

70. Exponential Probability Distribution
Time between
vehicle arrivals
at a toll booth
Time required
to complete
a questionnaire
Distance between
major defects
in a highway
SLOW

71. Exponential Probability Distribution
Density Function
for x > 0,  > 0
where:  = mean
e =

72. Exponential Probability Distribution
Cumulative Probabilities
where:
x0 = some specific value of x

73. Computing Probabilities for the Exponential Distribution
Example: The Schips loading dock example.
x = loading time and μ=15, which gives us
What is the probability that loading a truck will take between 6 minutes and 18 minutes?

74. Computing Probabilities for the Exponential Distribution
Since,
and
The probability that loading a truck will take between 6 minutes and 18 minutes is equal to =

75. Exponential Probability Distribution
The time between arrivals of
cars at Al’s full-service gas pump
follows an exponential probability
distribution with a mean time between arrivals of 3 minutes. Al would like to know the
probability that the time between two successive arrivals will be 2 minutes or less.
Example: Al’s Full-Service Pump

76. Exponential Probability Distribution
P(x < 2) = /3 = =
f(x) .1 .3 .4 .2 x
Time Between Successive Arrivals (mins.)

77. Exponential Probability Distribution
A property of the exponential distribution is that
the mean, m, and standard deviation, s, are equal.
Thus, the standard deviation, s, and variance, s 2,
for the time between arrivals at Al’s full-service
pump are:
s = m = minutes
s 2 = (3)2 = 9

78. Exponential Probability Distribution
The exponential distribution is skewed to the
right.
The skewness measure for the exponential
distribution is 2.

79. Relationship between the Poisson and Exponential Distributions
The Poisson distribution
provides an appropriate description
of the number of occurrences
per interval
The exponential distribution
provides an appropriate description
of the length of the interval
between occurrences