1. PROBABILITY DISTRIBUTIONS
KHATIJAHHUSNA BINTI ABD RANI
EQT271 SEM II 2014/2015
CHAPTER 1 PART 2
Slide was prepared by Miss Syafawati (with modification)

2. Probability Distributions
DISCRETE
CONTINUOUS
Binomial
Poisson
Normal

3. RANDOM VARIABLE A variable whose values are determined by chance
Random Process

4. RANDOM VARIABLE
DISCRETE
CONTINUOUS

5. Probability Distributions
A probability function is a function which assigns probabilities to the values of a random variable.
Individual probability values may be denoted by the symbol P(X=x), in the discrete case, which indicates that the random variable can have various specific values.
All the probabilities must be between 0 and 1;
0≤ P(X=x)≤ 1.
The sum of the probabilities of the outcomes must be 1.
∑ P(X=x)=1
It may also be denoted by the symbol f(x), in the continuous, which indicates that a mathematical function is involved.

6. X=number of “heads” after 3 flips a fair coin
Example 1
X=number of “heads” after 3 flips a fair coin
Probability
HHH
THH
HHT
THT
HTH
TTH
HTT
TTT

7. Check whether the distribution is a probability distribution. Solution
# so the distribution is not a probability distribution.
Example 2 X 1 2 3 4
P(X=x)
0.125
0.375
0.025

8. Binomial Distribution
An experiment in which satisfied the following characteristic is called a binomial experiment:
1. There must be a fixed number of trials.
2. Each trial can result in one of two outcomes, which we denote by success, S or failure, F.
3. The outcomes of each trial are independent.
4. The probability of success is constant from trial to trial, we denote the probability of success by p and the probability of failure is equal to (1 - p) = q.
Examples:
No. of getting a head in tossing a coin 10 times.
No. of getting a six in tossing 7 dice.
A firm bidding for contracts will either get a contract or not
Binomial Distribution
Bluman; Pg276

9. A binomial experiment consist of n identical trial with probability of success, p in each trial. The probability of x success in n trials is given by The Mean and Variance of X if X ~ B(n,p) are Mean : Variance : Std Deviation : where n is the total number of trials, p is the probability of success and q is the probability of failure.

10. Example 3

11. Solutions:

12. Cumulative Binomial Distribution
When the sample is relatively large, tables of Binomial are often used. Since the probabilities provided in the tables are in the cumulative form the following guidelines can be used:

13. Example 4
In a Binomial Distribution, n =12 and p = 0.3. Find the following probabilities.

14. Statistics; Formulae & Tables pg 13
Example 4 (a) 1 2 3 4 5 6 7 8 9 10 11 12
Statistics; Formulae & Tables pg 13

15. Statistics; Formulae & Tables pg 13
Example 4 (b) 1 2 3 4 5 6 7 8 9 10 11 12
Statistics; Formulae & Tables pg 13

16. Statistics; Formulae & Tables pg 13
Example 4 (c) 1 2 3 4 5 6 7 8 9 10 11 12
Statistics; Formulae & Tables pg 13

17. Statistics; Formulae & Tables pg 13
Example 4 (d) 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
Statistics; Formulae & Tables pg 13

18. Example 4 (e) 1 2 3 4 5 6 7 8 9 10 11 12

19. The number of CEOs who hold this view is 10.
Example 5
In August 2009, David and Maria conducted a survey for Fortune magazine to examine CEO`s attitudes toward employee`s personal problems. 30% of the CEOs interviewed felt that personal problems were none of the company`s business. Assume that this result is true for the current population of CEOs. Using the Binomial distribution tables, in a random samples of 15, find the probability that
The number of CEOs who hold this view is 10.
The number of CEOs who hold this view is between 9 and 12.
The number of CEOs who hold this view is at most 7.
Find the mean and standard deviation of binomial distribution.

20. Solution: i) ii) iii) iv)

21. Poisson distribution is the probability distribution of the number of successes in a given space*.
*space can be dimensions, place or time or combination of them
Used when n is large and p is small & when the independent variables occur over a period of time
Examples:
No. of cars passing a toll booth in one hour.
No. defects in a square meter of fabric
No. of network error experienced in a day.
Poisson Distribution

22. A random variable X has a Poisson distribution and it is referred to as a Poisson random variable if and only if its probability distribution is given by A random variable X having a Poisson distribution can also be written as

23. Example 6
Consider a Poisson random variable with . Calculate the following probabilities : a) Write the distribution of Poisson b) c) d)

24. Solution: a) b) c) d)

25. Example 7
Granma bakes chocolate chip cookies in batches 0f She puts 300 chips into the dough. When the cookies are done, she gives you one. What is the probability that your cookie contains no chocolate chips?

26. Poisson Approximation of Binomial Probailities
The Poisson distribution is suitable as an approximation of Binomial
probabilities when n is large and p is small. Approximation can be made
when , and either or

27. Example 8
Suppose a life insurance company insures the lives of men aged 42. If actuarial studies show the probability that any 42 year old man will die in a given year to be 0.001, find the exact probability that the company will have to pay x = 4 claims during a given year.

28. Solution:
Step:
Write the distribution of Binomial
The value for n is large and value of p is too small, check whether
If yes, proceed to solve using Poisson Approximation. Use formula

29. Statistics; Formulae & Tables pg 18

30. Read the following questions and decide whether the Poisson or the Binomial distribution should be used to answer it.
A typist makes on average 2 mistakes per page. What is the probability of a particular page having no errors on it?.
A computer crashes once every 2 days on average. What is the probability of there being 2 crashes in one week?
Components are packed in boxes of 20. The probability of a component being defective is 0.1. What is the probability of a box containing 2 defective components?

31. ICs are packaged in boxes of 10
ICs are packaged in boxes of 10. The probability of an ic being faulty is 2%. What is the probability of a box containing 2 faulty ics?
The mean number of faults in a new house is 8. What is the probability of buying a new house with exactly 1 fault?

32. Normal Probability Distribution
Numerous continuous variables have distribution closely resemble the normal distribution.
The normal distribution can be used to approximate various discrete probability distribution.
Normal Probability Distribution

33. CHARACTERISTICS OF NORMAL DISTRIBUTION ‘Bell Shaped’ Symmetrical
Mean, Median and Mode
are Equal
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

34. Many Normal Distributions
By varying the parameters μ and σ, we obtain different normal distributions

35. The Standard Normal Distribution
Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (Z)
Need to transform X units into Z units using
The standardized normal distribution (Z) has a mean of ,
and a standard deviation of 1,
Z is denoted by
Thus, its density function becomes

36. Calculating Probabilities for a General Normal Random Variable
Mostly, the probabilities involved x, a normal random variable with mean, and standard deviation,
Then, you have to standardized the interval of interest, writing it in terms of z, the standard normal random variable.
Once this is done, the probability of interest is the area that you find using the standard normal probability distribution.
Normal probability distribution,
Need to transform x to z using

37. Patterns for Finding Areas under the Standard Normal Curve

38. Example 9
Z table

39. Find the area under the standard normal curve of
Example 10
Find the area under the standard normal curve of 1

40. Exercise 1.6 Determine the probability or area for the portions of the Normal distribution described. Answer : a) , b) , c) , d) , e)

41. Example 11
Z table

42. Exercise 1.7

43. Solutions:
Z table

44. Suppose X is a normal distribution N(25,25). Find
Example 12
Suppose X is a normal distribution N(25,25). Find
Solutions

45. Exercise 1. 8 1. Suppose X is a normal distribution, N(70,4)
Exercise Suppose X is a normal distribution, N(70,4). Find a) b) 2. Suppose the test scores of 600 students are normally distributed with a mean of 76 and standard deviation of 8. The number of students scoring is from 70 to 82 is: Answer : 1. a) b) students

46. Normal Approximation of the Binomial Distribution
When the number of observations or trials n in a binomial experiment is relatively large, the normal probability distribution can be used to approximate binomial probabilities. A convenient rule is that such approximation is acceptable when

47. Continuous Correction Factor The continuous correction factor needs to be made when a continuous curve is being used to approximate discrete probability distributions. 0.5 is added or subtracted as a continuous correction factor according to the form of the probability statement as follows:

48. How do calculate Binomial Probabilities Using the Normal
Approximation?
Find the necessary values of n and p. Calculate
Write the probability you need in terms of x.
Correct the value of x with appropriate continuous correction factor (ccf).
Convert the necessary x-values to z-values using
Use Standard Normal Table to calculate the approximate probability.

49. Example 13
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.
Solutions:

50. Exercise 1.9
Suppose that 5% of the population over 70 years old has disease A. Suppose a random sample of 9600 people over 70 is taken. What is the probability that less than 500 of them have disease A?
Answer:

51. Normal Approximation of the Poisson Distribution
When the mean of a Poisson distribution is relatively large, the normal probability distribution can be used to approximate Poisson probabilities. A convenient rule is that such approximation is acceptable when

52. Example 14
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? Solution:

53. Exercise 1.10 The average number of accidental drowning in United States per year is 3.0 per population. Find the probability that in a city of population there will be less than 10 accidental drowning per year. Answer :

54. Exercise 1.11
Reported that the mean weekly income of a shift foreman in the glass industry is normally distributed with a mean of $1000 and standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is
Between $1000 and $1100.
Between $790 and $1000.
Between $840 and $1200.
Answer : a) , b) , c)