2. Free Powerpoint Templates ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS

3. PROBABILITY DISTRIBUTION CHAPTER 3

4. PROBABILITY DISTRIBUTION 3.1 Introduction 3.2 Binomial distribution 3.3 Poisson distribution 3.4 Normal distribution

5. 3.1 INTRODUCTION A probability distribution is obtained when probability values are assigned to all possible numerical 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.

6. It may also be denoted by the symbol f(x), in the continuous, which indicates that a mathematical function is involved. The sum of the probabilities for all the possible numerical events must equal 1.0.

7. 3.2 THE BINOMIAL DISTRIBUTION Definition 3.1 : 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.

8. Definition 3.2 : 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  x = 0, 1, 2,......, n

9. Definition 3.3 :The Mean and Variance of X If X ~ B(n,p), then where  n is the total number of trials,  p is the probability of success and  q is the probability of failure. MeanVariance Standard deviation

10. E XAMPLE 3.1

11. SOLUTIONS

13. 3.3 The Poisson Distribution Definition 3.4  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

14.  λ (Greek lambda) is the long run mean number of events for the specific time or space dimension of interest.  A random variable X having a Poisson distribution can also be written as

15. E XAMPLE 3.2 Given that, find

16. SOLUTIONS

17. E XAMPLE 3.3 Suppose that the number of errors in a piece of software has a Poisson distribution with parameter. Find a) the probability that a piece of software has no errors. b) the probability that there are three or more errors in piece of software. c) the mean and variance in the number of errors.

18. SOLUTIONS

19. 3.4 The Normal Distribution Definition 3.5

20. The Standard Normal 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.

21. Standardizing A Normal Distribution

22. E XAMPLE 3.4 Determine the probability or area for the portions of the Normal distribution described. (using the table)

23. SOLUTIONS

25. E XAMPLE 3.5

26. SOLUTIONS

27. E XAMPLE 3.6 Suppose X is a normal distribution N(25,25). Find

28. SOLUTIONS

29. 3.4.1 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

30. Definiton 3.6

31. 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:

32. Example 3.7 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.

33. Solutions

34. 3.4.1 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  Definition 3.7

35. Example 3.8 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?

36. Solutions