1. Chapter 6 Introduction to Continuous Probability Distributions
Business Statistics:
A Decision-Making Approach
8th Edition
Chapter 6 Introduction to Continuous Probability Distributions

2. Chapter Goals After completing this chapter, you should be able to:
Convert values from any normal distribution to a standardized z-score
Find probabilities using a normal distribution table
Apply the normal distribution to business problems
Recognize when to apply the uniform and exponential distributions (not covered)

3. Continuous Probability Distributions
A continuous probability distribution differs from a discrete probability distribution in several ways.
A continuous random variable can take on an infinite number of values. That is, the exact probability for a specific variable is always zero.
As a result, a continuous probability distribution cannot be expressed in tabular form.
Instead, an equation or formula is used to describe a continuous probability distribution.

4. Probability Density Function
The equation for describing a continuous probability distribution is called a probability density function and noted as “f(x).”
Sometimes refer to as Density function
pdf or PDF

5. Probability Density Function Example
The shaded area in the graph represents the probability that the random variable X is less than or equal to a.
Cumulative probability
Always to the left
The probability that the random variable X  is exactly equal to a would be zero.

6. Types of Continuous Distributions
Three types
Normal
Uniform (so easy…by yourself)
Exponential (not cover)

7. The Normal Distribution
‘Bell Shaped’
Symmetrical
Mean=Median=Mode
Location is determined by the
mean, μ
Spread is determined by the standard deviation, σ
The random variable has an infinite theoretical range:
+  to  
f(x) σ x μ
Mean
Median
Mode

8. Position and Shape of the Normal Curve

9. Basis for the empirical rules
68.26% of the area under the curve falls within +/- 1 Stdev.
95.44% of the area under the curve falls within +/- 2 Stdev.
99.73% of the area under the curve falls within +/- 3 Stdev.

10. Probability as Area Under the Curve
The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below
f(x)
0.5
0.5 μ x

11. Normal Probability Distribution
The normal distribution refers to a family of continuous probability distributions: same equation.
Where:  = mean
 = standard deviation
 =
e =

12. The Standard Normal Distribution
Also known as the “z” distribution
Mean is defined to be 0
Standard Deviation is 1
f(z) 1 z
Values above the mean have positive z-values Values below the mean have negative z-values

13. Translation to the Standard Normal Distribution
Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation:
z is the number of standard deviations units that
x is away from the population mean

14. Example 1
If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is
This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100.

15. Comparing x and z units 100 250 x 3.0 z μ = 100 σ = 50
3.0 z
Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z)

16. Why standardizing?
To measure variable with different means and/or standard deviations on a single scale.
Statistically (based standard normal distribution), are they really or fundamentally different matters?
SAT mean 1020 with standard deviation of 160. You scored What is your percentile?
Average adult male height is 70 inches with standard deviation of 2 inches. You are 73 inches tall. What is your height percentile?

17. Z-scores for Two Problems
Both are 1.5 Std Dev above the mean….
Download the Normal Dist. using Excel file from the class website and then select the first tab “Example 1”….

18. Four Excel Functions NORMDIST(x, mean, standard_dev, cumulative)
Gives the probability that a number falls at or below a given value of a normal distribution
NORMSDIST(z)
Translates the number of standard deviations (z) into cumulative probabilities

19. Four Excel Functions
The distribution of heights of American women aged 18 to 24 is approximately normally distributed with a mean of 65.5 inches ( cm) and a standard deviation of 2.5 inches (6.35 cm).
What percentage of these women is taller than 5' 8", that is, 68 inches ( cm)?
NORMDIST(68, 65.5, 2.5, TRUE)  =  84.13%
STANDARDIZE(68, 65.5, 2.5) = 1, NORMSDIST(1) = 84.13%
Cumulative probability (to the left on the normal curve)
Thus, % = 15.87% or NORMSDIST(-1) = 15.87%

20. Four Excel Functions NORMINV(probability, mean, standard_dev)
Calculates the x variable given a probability
NORMSINV(probability)
Given the probability that a variable is within a certain distance of the mean, it finds the z value

21. Four Excel Functions
How tall would a woman need to be if she wanted to be among the tallest 75% of American women (find 25% shorter instead as the right figure shows)?
NORMINV(0.25, 65.5, 2.5)  = inches
Find the z values that mark the boundary that is 25% less than the mean and 25% (75%) more than the mean.
NORMSINV(0.25) =
NORMSINV(0.75) = 0.674

22. Formula Table We have X Z p We want X=μ+zσ Norminv(p,μ,σ) Normsinv(p)
Normdist(x,μ,σ,1)
Normsdist(z)

23. Example 1
An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days?

24. Example 2
Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110?

25. Example 3
Molly earned a score of 940 on a national achievement test. The mean test score was 850 with a standard deviation of 100. What proportion of students had a higher score than Molly? (Assume that test scores are normally distributed.)

26. The Standard Normal Table
The Standard Normal table in the textbook (Appendix D)
Gives the probability from the mean (zero)
up to a desired value for z
0.4772
Example:
P(0 < z < 2.00) = z
2.00

27. The Standard Normal Table
(continued)
The Standard Normal Table gives the probability between the mean and a certain z value
The z value ALWAYS refers to the area between some value (-z or +z) and the mean
Since the distribution is symmetrical, the Standard Normal Table only displays probabilities for ½ of the full distribution

28. The Standard Normal Table
(continued)
The column gives the value of z to the second decimal point
The row shows the value of z to the first decimal point
The value within the table gives the probability from z = 0 up to the desired z value .
2.0
.4772
P(0 < z < 2.00) =
2.0

29. General Procedure for Finding Probabilities
Determine m and s
Define the event of interest
e.g., P(x > x1)
Convert to standard normal
Use the table to find the probability

30. z Table Example
Suppose x is normal with mean 8.0 and standard deviation Find P(8 < x < 8.6)
Calculate z-values: 8
8.6 x
0.12 Z
P(8 < x < 8.6)
= P(0 < z < 0.12)

31. z Table Example
(continued)
Suppose x is normal with mean 8.0 and standard deviation Find P(8 < x < 8.6)
= 8
 = 5
= 0
 = 1 x z 8
8.6
0.12
P(8 < x < 8.6)
P(0 < z < 0.12)

32. Solution: Finding P(0 < z < 0.12)
Standard Normal Probability
Table (Portion)
P(8 < x < 8.6)
= P(0 < z < 0.12)
.02 z
.00
.01
0.0478
0.0
.0000
.0040
.0080
0.1
.0398
.0438
.0478
0.2
.0793
.0832
.0871 Z
0.3
.1179
.1217
.1255
0.00
0.12

33. Finding Normal Probabilities
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(x < 8.6)
The probability of obtaining a value less than 8.6
P = 0.5 Z
8.0
8.6

34. Finding Normal Probabilities
(continued)
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(x < 8.6)
0.0478
0.5000
P(x < 8.6)
= P(z < 0.12)
= P(z < 0) + P(0 < z < 0.12)
= = Z
0.00
0.12

35. Upper Tail Probabilities
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(x > 8.6) Z
8.0
8.6

36. Upper Tail Probabilities
(continued)
Now Find P(x > 8.6)…
P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12)
= =
0.0478
0.5000
0.4522 Z Z
0.12
0.12

37. Lower Tail Probabilities
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(7.4 < x < 8) Z
8.0
7.4

38. Lower Tail Probabilities
(continued)
Now Find P(7.4 < x < 8)…the probability between 7.4 and the mean of 8
The Normal distribution is symmetric, so we use the same table even if z-values are negative:
P(7.4 < x < 8)
= P(-0.12 < z < 0) =
0.0478 Z
8.0
7.4

39. Recall Tchebyshev from Chpt. 3
Empirical Rules
What can we say about the distribution of values around the mean if the distribution is normal?
f(x)
μ ± 1σ covers about 68% of x’s σ σ
Recall Tchebyshev from Chpt. 3 x
μ-1σ μ
μ+1σ
68.26%

40. μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s
The Empirical Rule
(continued)
μ ± 2σ covers about 95% of x’s
μ ± 3σ covers about 99.7% of x’s 3σ 3σ 2σ 2σ μ x μ x
95.44%
99.72%

41. Importance of the Rule If a value is about 2 or more standard
deviations away from the mean in a normal
distribution, then it is far from the mean
The chance that a value that far or farther
away from the mean is highly unlikely, given
that particular mean and standard deviation

42. Normal Probabilities in PHStat
We can use Excel and PHStat to quickly
generate probabilities for any normal
distribution
We will find P(8 < x < 8.6) when x is
normally distributed with mean 8 and
standard deviation 5

43. Select desired options and enter values
PHStat Dialogue Box
Select desired options and enter values

44. PHStat Output

45. The Uniform Distribution
The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable
Referred to as the distribution of “little information”
Probability is the same for ANY interval of the same width
Useful when you have limited information about how the data “behaves” (e.g., is it skewed left?)

46. The Uniform Distribution
(continued)
The Continuous Uniform Distribution:
f(x) =
where
f(x) = value of the density function at any x value
a = lower limit of the interval of interest
b = upper limit of the interval of interest

47. The Mean and Standard Deviation for the Uniform Distribution
The mean (expected value) is:
The standard deviation is
where
a = lower limit of the interval from a to b
b = upper limit of the interval from a to b

48. Steps for Using the Uniform Distribution
Define the density function
Define the event of interest
Calculate the required probability
f(x) x

49. Uniform Distribution Example: Uniform Probability Distribution
Over the range 2 ≤ x ≤ 6: 1
f(x) = = for 2 ≤ x ≤ 6
6 - 2
f(x)
.25 x 2 6

50. Uniform Distribution Example: Uniform Probability Distribution
Over the range 2 ≤ x ≤ 6:

51. The Exponential Distribution
Used to measure the time that elapses between two occurrences of an event (the time between arrivals)
Examples:
Time between trucks arriving at a dock
Time between transactions at an ATM Machine
Time between phone calls to the main operator
Recall l = mean for Poisson (see Chpt. 5)

52. The Exponential Distribution
The probability that an arrival time is equal to or less than some specified time a is
where 1/ is the mean time between events and e =
NOTE: If the number of occurrences per time period is Poisson with mean , then the time between occurrences is exponential with mean time 1/  and the standard deviation also is 1/l

53. Exponential Distribution
(continued)
Shape of the exponential distribution
f(x)
 = 3.0
(mean = .333)
 = 1.0
(mean = 1.0)
= 0.5
(mean = 2.0) x

54. Example
Example: Customers arrive at the claims counter at the rate of 15 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes?
Time between arrivals is exponentially distributed with mean time between arrivals of 4 minutes (15 per 60 minutes, on average)
1/ = 4.0, so  = .25
P(x < 5) = 1 - e-a = 1 – e-(.25)(5) =
There is a 71.35% chance that the arrival time between consecutive customers is less than 5 minutes

55. Using PHStat

56. Chapter Summary Reviewed key continuous distributions
normal
uniform
exponential
Found probabilities using formulas and tables
Recognized when to apply different distributions
Applied distributions to decision problems

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Printed in the United States of America.