1. Binomial Applet

2. Continuous Probability Distributions
Chapter 6
Continuous Probability Distributions

3. Discrete vs. Continuous Random Variables
For discrete random variables we find the probability the variable will take on a specific value. For continuous random variable we find the probability the variable will fall in some range.
For discrete random variables we find the probability associated with some value using a probability function. For a continuous random variable we find the area under a curve between two points.

4. Probability Density Curves
The area under the curve is equal to 1
The area under the curve between two values of the variable measures the probability the value will fall between those values
A probability density curve is defined by a probability density function

5. Uniform Probability Distribution
The probability of the variable falling into some interval is the same for all equally-sized intervals.

6. Uniform Probability Density Function

7. Mean and Variance of a Uniform Distribution

8. Uniform Distribution, Example
Assume that a delivery truck takes between 200 minutes and 250 minutes to complete its delivery route. Assume that the amount of time taken is uniformly distributed.
What is the probability the truck will take between 210 and 230 minutes?

9. Uniform Distribution, Example
What is the probability the truck will take between 210 and 230 minutes?
(230 – 210)/(250 – 200) = 20/50 = 0.4

10. Uniform Distribution, Example
What is the probability the truck will take more than 240 minutes?

11. Uniform Distribution, Example
What is the probability the truck will take more than 240 minutes?
(250 – 240)/(250 – 200) = 10/50 = 0.2

12. Uniform Distribution, Example
What is the probability the truck will take less than 220 minutes?

13. Uniform Distribution, Example
What is the probability the truck will take less than 220 minutes?
(220 – 200)/(250 – 200) = 20/50 = 0.4

14. Uniform Distribution, Example
What is the mean and variance of this distribution?
E(x) = ( )/2 = 225
Var(x) = (250 – 200)2/12 = 2500/12 =

15. Importance of the Normal Distribution
Many variables have a distribution similar to that of the normal distribution
The means of samples are normally distributed (given samples of 30 or more)

16. Characteristics of the Normal Distribution
Symmetrical and bell-shaped
The tails of the distribution asymptotically converge on the x axis
The location of the distribution on the x axis is determined by the mean of the distribution
The shape of the distribution is determined by the variance
The total area under the curve is equal to 1, the areas to either side of the mean equal 0.5
Moving out a given number of standard deviations will always capture the same share of the distribution

17. Normal Distributions
Wikipedia

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

19. Standard Normal Distribution
= 0
s=1
To convert a normal distribution to standard normal:
z = (x - m)/s

20. Characteristics of the Normal Distribution

21. Normal Probability Density Function

22. Standard Normal Table

23. Standard Normal Table

24. Standard Normal Table

25. Normal Distribution, Exercises
Find: P(z < 2)
P(z < 2) = .9772

26. Normal Distribution, Exercises
Find: P(z < -2)
P(z < -2) = .0228

27. Normal Distribution, Exercises
Find: P(0 < z < 2)
P(0 < z < 2) = = .4772 Or
P(0 < z < 2) = = .4772

28. Normal Distribution, Exercises
Find: P(-1 < z < 0)
P(-1 < z < 0) = P(0 < z < 1) = = .3413 Or
P(-1 < z < 0) = = .3413

29. Normal Distribution, Exercises
Find: P(z > -1)
P(z > -1) = P(z < 1) = .8413 Or
P(z > -1) = = .8413

30. Normal Distribution, Exercises
Find: P(z > 2)
P(z > 2) = = .0228 Or
P(z > 2) = P(z < -2) = .0228

31. Normal Distribution, Exercises
Find: P(1 < z < 1.5)
P(1 < z < 1.5) = = Or
P(1 < z < 1.5) = P(-1.5 < z < -1) = = .0919

32. Normal Distribution, Exercises
Find: P(-1.5 < z < -0.5)
P(-1.5 < z < -0.5) = P(0.5 > z > 1.5) = = .2417 Or
P(-1.5 < z < -0.5) = = .2417

33. Normal Distribution, Exercises
Find: P(-1 < z < 2)
P(-1 < z < 2) = =

34. Standard Normal Probability Distribution, Example
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.
Pep
Zone
5w-20
Motor Oil

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

36. Standard Normal Probability Distribution, Example
Solve for the z score:
Z = (x – m)/s
= (20 – 15)/6
= .83
Look up value in standard normal table

37. Standard Normal Probability Distribution, Example
Pep
Zone
5w-20
Motor Oil
P(z < .83)

38. Standard Normal Probability Distribution, Example
Compute the area under the curve to the right of 0.83
P(z > .83) = 1 – P(z < .83) = =

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

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

41. Standard Normal Probability Distribution, Example
Pep
Zone
5w-20
Motor Oil
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)

42. Standard Normal Probability Distribution, Example
Pep
Zone
5w-20
Motor Oil
Step 2: Convert z.05 to the corresponding value of x.
x = m+z.05s
= (6)
= or 25
A reorder point of 25 gallons will place the
probability of a stockout during lead-time at
(slightly less than) .05.

43. Standard Normal Probability Distribution, Example
Pep
Zone
5w-20
Motor Oil
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.

44. Practice Homework
P. 241, #12, 20
P. 246, #30