1. Normal Distribution

2. Before Starting Normal Distribution
P(8<x < 12) = ? x
f (x ) 5 15 12
1/10
P( < 12) = ?
P( < 8) = ? x
f (x ) 5 15 12
1/10
P(8<x < 12) = = .4

3. The Normal Probability Distribution
Graph of the Normal Probability Density Function
f (x ) x 

4. The Normal Curve
The shape of the normal curve is often illustrated as a bell-shaped curve.
The highest point on the normal curve is at the mean of the distribution.
The normal curve is symmetric.
The standard deviation determines the width of the curve.

5. The Normal Curve
The total area under the curve the same as any other probability distribution is 1.
The probability of the normal random variable assuming a specific value the same as any other continuous probability distribution is 0.
Probabilities for the normal random variable are given by areas under the curve.

6. The Normal Probability Density Function
where
 = mean
 = standard deviation
 =
e =

7. The Standard Normal Probability Density Function
where
 = 0
 = 1
 =
e =

8. Given any positive value for z, the table will give us the following probability
The table will give this probability
Given positive z
The probability that we find using the table is the probability of having a standard normal variable between 0 and the given positive z.

9. Given z find the probability

10. Given any probability between 0 and
Given any probability between 0 and .5,, the table will give us the following positive z value
Given this probability between 0 and .5
The table will give us
this positive z

11. Given the probability find z find

12. What is the z value where probability of a standard normal variable to be greater than z is .1
10%
40%

13. The End

14. Standard Normal Probability Distribution (Z Distribution)

15. Standard Normal Probability Distribution
A random variable that has a normal distribution with a mean of zero and a standard deviation of one is said to have a standard normal probability distribution.
The letter z is commonly used to designate this normal random variable.
The following expression convert any Normal Distribution into the Standard Normal Distribution

16. Example: Pep Zone
Pep Zone sells auto parts and supplies including multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed.
The store manager is concerned that sales are being lost due to stockouts while waiting for an order.
It has been determined that leadtime demand is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons.
In Summary; we have a N (15, 6): A normal random
variable with mean of 15 and std of 6.
The manager would like to know the probability of a stockout, P(x > 20).

17. Standard Normal Distribution
z = (x -  )/
= ( )/6
= .83
Area = .5 z
.83

18. Example: Pep Zone

19. The Probability of Demand Exceeding 20
.83
Area = .2967
Area = .5
Area = .2033 z
The Standard Normal table shows an area of for the region between the z = 0 line and the z = .83 line above. The shaded tail area is = The probability of a stockout is

20. Example: Pep Zone
If the manager of Pep Zone wants the probability of a
stockout to be no more than .05, what should the
reorder point be?
Let z.05 represent the z value cutting the tail area of .05.
Area = .05
Area = .5
Area = .45
z.05

21. Example: Pep Zone Using the Standard Normal Probability Table
We now look-up the area in the Standard Normal Probability table to find the corresponding z.05 value. z.05 = is a reasonable estimate.

22. Example: Pep Zone The corresponding value of x is given by
x =  + z.05
= (6)
= 24.87
A reorder point of gallons will place the
probability of a stockout during leadtime at .05.
Perhaps Pep Zone should set the reorder point at 25 gallons to keep the probability under .05.

23. Example: Aptitude Test
A firm has assumed that the distribution of the aptitude test of people applying for a job in this firm is normal.
The following sample is available.
79 84

24. Example: Mean and Standard Deviation
We first need to estimate mean and standard deviation

25. z Values
What test mark has the property of having 10% of test marks being less than or equal to it
To answer this question, we should first answer the following
What is the standard normal value (z value), such that 10% of z values are less than or equal to it?
10%

26. z Values We need to use standard Normal distribution in Table 1. 10%

27. z Values
10%
40%

28. z Values
40%
z = 1.28
10%
z =

29. z Values and x Values
The standard normal value (z value), such that 10% of z values are less than or equal to it is z = -1.28
To transform this standard normal value to a similar value in our example, we use the following relationship
The normal value of test marks such that 10% of random variables are less than it is 55.1.

30. z Values and x Values
Following the same procedure, we could find z values for cases where 20%, 30%, 40%, …of random variables are less than these values. Following the same procedure, we could transform z values into x values.
Lower 10%
Lower 20%
Lower 30%
Lower 40%
Lower 50%
Lower 60%

31. Example : Victor Computers
Victor Computers manufactures and sells a
general purpose microcomputer. As part of a study to evaluate sales personnel, management wants to determine if the annual sales volume (number of units sold by a salesperson) follows a normal probability distribution.
A simple random sample of 30 of the salespeople was taken and their numbers of units sold are below.
(mean = 71, standard deviation = 18.54)
Partition this Normal distribution into 6 equal probability parts

32. Example : Victor Computers
Areas
= 1.00/6
= .1667
53.02 71
88.98 = (18.54)
63.03
78.97

33. The End

34. 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.
Set  = np
Add and subtract 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).

35. The Exponential Probability Distribution
Exponential Probability Density Function
for x > 0,  > 0
where  = mean
e =
Cumulative Exponential Distribution Function
where x0 = some specific value of x

36. Example: Al’s Carwash
The time between arrivals of cars at Al’s Carwash
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.
P(x < 2) = /3 = = .4866

37. Example: Al’s Carwash Graph of the Probability Density Function F (x ).1 .3 .4 .2
P(x < 2) = area = .4866

38. The End