1. Continuous Probability Distributions
l Chapter 6 l
Continuous Probability Distributions
6.1 Normal Probability Distribution
6.2 Standard Normal Probability
Distribution
6.3 Binomial Approximation
6.4 Poisson Approximation

2. Introduction

3. 6.1 Normal Probability Distribution

4. 6.1 Normal Probability Distribution
The Normal Distribution has:
mean = median = mode
symmetry about the center
50% of values less than the mean and 50% greater than the mean
Location is determined by the mean, μ
Spread is determined by the standard deviation, σ
The random variable has an infinite theoretical range: to

5. 6.1 Normal Probability Distribution
By varying the parameters μ and σ, we obtain different normal distributions

6. 6.1 Normal Probability Distribution
Applications of normal distribution
Many naturally occurring random processes tend to have a distribution that is approximately normal.  Examples can be found in any field, these include:
heights and weights of adults
length and width of leaves of the same species
actual weights of rice in 5 kg bags sold in supermarkets

7. 6.2 Standard Normal Probability Distribution
The normal distribution with parameters and is called a standard normal distribution.
A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by
Why standardize?
It also makes life easier because we only need one table (the Standard Normal Distribution Table) to fine probability rather than doing calculations individually for each value of mean and standard deviation.

8. 6.2 Standard Normal Probability Distribution
Why standardize?
Standardized continuous data
Can be used to find probability using the same table (Standard Normal Distribution Table)
Standardize using formula
Standard Normal
Distribution Table
Unstandardized continuous data
Need to be calculated independently to find probability

9. 6.2 Standard Normal Probability Distribution
Standardizing A Normal Distribution
If is a normal random variable with and , the random variable
is a normal random variable with and
That is is a standard normal variable

10. 6.2 Standard Normal Probability Distribution
Example

11. 6.2 Standard Normal Probability Distribution
Example: Professor Willoughby is marking a test.
Here are the students results (out of 60 points):
20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17
Most students didn't even get 30 out of 60, and most will fail, so, to avoid that the Prof decides to STANDARDIZE all the scores and only fail people 1 standard deviation below the mean.
The Mean is 23, and the Standard Deviation is 6.6, and these are the Standard Scores:
Only 2 students will fail (the ones who scored 15 and 14 on the test)
True
value 20 15 26 32 18 28 35 14 22 17
Standardized
-0.45
-1.21
0.45
1.36
-0.76
0.76
1.82
-1.36
-0.15
-0.91

12. 6.2 Standard Normal Probability Distribution
Patterns for finding areas under the Standard Normal Curve

13. 6.2 Standard Normal Probability Distribution
Example

14. 6.2 Standard Normal Probability Distribution
Exercise

15. 6.2 Standard Normal Probability Distribution
Solution

16. 6.2 Standard Normal Probability Distribution
Solution

17. 6.2 Standard Normal Probability Distribution

18. 6.2 Standard Normal Probability Distribution

19. 6.2 Standard Normal Probability Distribution
Calculate the probabilities using calculator
Mode: SD 1
Shift 3
P(1) for
Q (2) for
R (3) for

20. 6.2 Standard Normal Probability Distribution
Extra exercise

21. 6.2 Standard Normal Probability Distribution
Exercise (Application)
The masses of a well known brand of breakfast cereal are normally distributed with mean of 250g and standard deviation of 4g. Find the probability of a packet containing more than 254.4g.

22. 6.2 Standard Normal Probability Distribution
Solution
Let X be the random variable “masses of cereal in grams” where X~N(250, 16).
There is 13.57% probability that a packet of breakfast cereal contain more than 254.4g weight.

23. 6.2 Standard Normal Probability Distribution
Extra exercise (Application)
A battery has a lifetimes which are normally distributed with a mean of 62 hours and a standard deviation of 3 hours. What is the probability of battery lasting less than 68 hours?

24. 6.2 Standard Normal Probability Distribution
Extra exercise (Application)
A carton of orange juice has a volume which is normally distributed with a mean of 120ml and a standard deviation of 1.8ml. Find the probability that the volume is more than 118ml.
The pulse rate is a measure of the number of heart beats per minute. Suppose that the pulse rates for adults are assumed to be normally distributed with a mean of 78 and a standard deviation of 12. Find the probability that adults will have the pulse rates between 60 and 100.

25. 6.2 Standard Normal Probability Distribution
Finding the value of
Upper tail probabilities or areas of the distribution of Z have also been tabulated. An entry in the table specifies a value of such that an area lies to its right. In other words,

26. 6.2 Standard Normal Probability Distribution
Solution

27. 6.2 Standard Normal Probability Distribution
Solution

28. 6.2 Standard Normal Probability Distribution
Solution - + - +

29. 6.2 Standard Normal Probability Distribution
Example
In January 2003, the American worker spent an average of 77 hours logged on to the internet while at work. Assume that the population mean is 77 hours, the times are normally distributed, and the standard deviation is 20 hours. A person is classified as heavy user if he or she is in the upper 20% of usage. How many hours did a worker have to be logged on to be considered a heavy user?

30. 6.2 Standard Normal Probability Distribution
Solution
Let X be the random variable “hours of worker spent on internet” where
A worker have to be logged on hours while at work to be considered as a heavy user

31. 6.3 Binomial Approximation
Consist of two kind of approximation
Binomial approximation to Poisson distribution
Binomial approximation to Normal distribution
Whenever n for any Binomial distribution is ≥ 30, Binomial distribution technique is ruled out and will be substituted with either Poisson or Normal distribution technique. The approximation guideline are as followed:
Approximation guideline
Binomial to n np nq
Distribution
Poisson
≥ 30
< 5 -
Normal
≥ 5

32. 6.3 Binomial Approximation
Binomial approximation to Poisson distribution
Example
Suppose a life insurance company insures the lives of 4000 men aged 42. If actual studies show that the probability that any 42-year-old man will die in a given year to be (0.1%), find the probability that the company will have to pay more than 10 claims due to death during a given year.

33. 6.3 Binomial Approximation
Binomial approximation to Poisson distribution
Solution
Let X = number of claims/number of any 42-year-old man will die in a given year
Where X~B(4000,0.001)
Use Poisson approximation since
There are 0.28% probability that the company will have to pay more than 10 claims due to death during a given year
Look at Poisson table!

34. 6.3 Binomial Approximation
Binomial approximation to Poisson distribution
Exercise
The percentage that a person will develop an infection even after taking a vaccine that was supposed to prevent the infection is 0.3%. In a random sample of 200 people in a community who got the vaccine, what is the probability that two or fewer people will be infected?

35. 6.3 Binomial Approximation
Binomial approximation to Normal distribution

36. 6.3 Binomial Approximation
Binomial approximation to Normal distribution
Continuous Correction Factor (c.c)
Continuous correction factor need to be made when a continuous curve is being used to approximate discrete probability distributions.

37. 6.3 Binomial Approximation
Binomial approximation to Normal distribution
Example
In a certain country, 45% of registered voters are male. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males.

38. 6.3 Binomial Approximation
Binomial approximation to Normal distribution
Solution
Let X be the random variable “number of male voters” where X~B (300, 0.45).

39. 6.3 Binomial Approximation
Binomial approximation to Normal distribution
Solution
There are 1.19% probability that out of 300 registered voters, at least 155 voters are males

40. 6.3 Binomial Approximation
Binomial approximation to Normal distribution
Exercise
According to a survey by Duit magazine, 27% of women expect to support their parents financially. Assume that this percentage holds true for the current population of all women. Suppose that a random sample of 300 women is taken. Find the probability that exactly 79 of the women in this sample expect to support their parents financially.
Aonang Beach Resort Hotel has 120 rooms. In the spring months, hotel room occupancy is approximately 75%.
What is the probability that 100 or more rooms are occupied on a given day.
What is the probability that 80 or fewer rooms are occupied on a given day?

41. 6.4 Poisson Approximation
Poisson distribution can only be approximated to Normal distribution
It is applied when the mean of a Poisson distribution is relatively large.
A convenient rule is that such approximation is acceptable when

42. 6.4 Poisson Approximation
Example
A grocery store has an ATM machine inside. An average of 5 customers per hour comes to use the machine. What is the probability that more than 30 customers come to use the machine between 8.00 am and 5.00 pm?

43. 6.4 Poisson Approximation
Solution
Let X be the random variable “number of customers per hour” where
Let X be the r.v. “number of customers for 9 hours” where

44. 6.4 Poisson Approximation
Solution
There are 98.46% that more than 30 customers come to use the machine between 8.00 am and 5.00 pm
-2.16

45. 6.4 Poisson Approximation
Exercise
In a university, the average of the students that come to the student health center is 5 students per hour. What is the probability that at least 40 students will come to the student health center from 9.00 am to 6.00 pm?
Suppose that at a certain automobile plant the average number of work stoppages per day due to equipment problems during the production process is 12. What is the approximate probability of having 15 or fewer work stoppages due to equipment problems on any given day?