1.  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. Probability Distributions

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

3. 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: 1. No. of getting a head in tossing a coin 10 times. 2. No. of getting a six in tossing 7 dice. 3. A firm bidding for contracts will either get a contract or not

4.  Check whether the distribution is a probability distribution.  Solution  # so the distribution is not a probability distribution. X01234 P(X=x)0.1250.3750.0250.3750.125

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.

7. Bin. table

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

10. 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 (i) The number of CEOs who hold this view is 10. (ii) The number of CEOs who hold this view is between 9 to 12. (iii) The number of CEOs who hold this view is at most 7. (iv) Find the mean and standard deviation of binomial distribution.

11.  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: 1. No. of cars passing a toll booth in one hour. 2. No. defects in a square meter of fabric 3. 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. Given that, fin 0.0307

14. 16 15

15.  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. 1. Given that Find (ans: 0.36, 0.16, 1.0, 0.64, 0.8, 0.48). 2. 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: 0.2668) ii) what is the probability that at least three workers take public transportation daily? (ans: 0.6172)

17. 3. Let Using Poisson distribution table, find i) (ans: 0.1550, 0.0655) ii) (ans: 0.9977, 0.9924) iii) (ans: 0.7697) 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 0.002. Compute the probability that: i) At least 5 watches will need to warranty work. (ans: 0.0527) ii) At most than 3 watches will need warranty work. (ans: 0.8571) iii) Less than 7 watches will need warranty work. (ans: 0.9955)

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

19. 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   Mean = Median = Mode X f(X) μ σ

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

23. a) Find the area under the standard normal curve of a) Find the area under the standard normal curve of

24. Z table

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

26. Z table

30. Suppose X is a normal distribution N(25,25). Find Solutions 0.5+0.3413 = 0.8413

31. 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.

33.  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. 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. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males. Solutions:

36. 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.  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. 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. 1. 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 a) Between $1000 and $1100. b) Between $790 and $1000. c) 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. a) 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? b) 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?