1. 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.

2. Probability Distributions
Discrete
Probability Distributions
Continuous
Probability Distributions
Binomial
Normal
Poisson

3. Binomial Distribution
An experiment in which satisfied the following characteristic is called a binomial experiment:
1. The random experiment consists of n identical trials.
2. Each trial can result in one of two outcomes, which we denote by success, S or failure, F.
3. The trials 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

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

5. 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.

6. Example

7. Solutions:
Bin. table

8. 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:

9. Example a) b) c) d) e) Bin. table
In a Binomial Distribution, n =12 and p = 0.3. Find the following probabilities. a) b) c) d) e)
Bin. table

10. EXAMPLE
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 to 12.
The number of CEOs who hold this view is at most 7.
Find the mean and standard deviation of binomial distribution.

11. The Poisson Distribution
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
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.

12. 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

13. Example :
Given that , fin
0.0307

14. Example : 16 15

15. Poisson Approximation of Binomial Probabilities
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
Example:
0.9786

16. EXERCISE 1 Given that Find (ans: 0.36, 0.16, 1.0, 0.64, 0.8, 0.48).
In Kuala Lumpur, 30% of workers take public transportation. In a sample of 10 workers,
i) what is the probability that exactly three workers take public transportation daily? (ans: )
ii) what is the probability that at least three workers take public transportation daily? (ans: )

17. 3. Let Using Poisson distribution table, find i) (ans: 0. 1550, 0
3. Let Using Poisson distribution table, find i) (ans: , ) ii) (ans: , ) iii) (ans: ) 4. Last month ABC company sold 1000 new watches. Past experience indicates that the probability that a new watch will need repair during its warranty period is Compute the probability that: i) At least 5 watches will need to warranty work. (ans: ) ii) At most than 3 watches will need warranty work. (ans: ) iii) Less than 7 watches will need warranty work. (ans: )

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

19. The Normal Distribution
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

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

21. 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 0, and a standard deviation of 1,
Z is denoted by
Thus, its density function becomes

22. Patterns for Finding Areas under the Standard Normal Curve

23. EXAMPLE
Find the area under the standard normal curve of

24. Example
Z table

25. Exercises
Determine the probability or area for the portions of the Normal distribution described.

26. Solutions:
Z table

27. Example
Z table

28. Exercises

29. Solutions
Z table

30. Example (Transform X into Z)
Suppose X is a normal distribution N(25,25). Find
Solutions =

31. EXERCISES 2:
1. 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 scoring is from 70 to 82.

32. Example

33. 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

34. 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:

35. Example In a certain country, 45% of registered voters are male
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.
Solutions:

36. Exercises
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 fewer than 500 of them have disease A?
Answer:

37. 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

38. Example & solution:
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?

39. Exercise 3 :
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.

40. 2. A study by Great Southern Home Insurance revealed that none of the stolen goods were recovered by the homeowners in 80 percent of reported thefts.
During a period in which 200 thefts occurred, what is the probability that no stolen goods were recovered in 170 and more of the robberies?
During a period in which 200 thefts occurred, what is the probability that no stolen goods were recovered in at least 150 of the robberies?