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 (PART 1) 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. Probability distribution can be classified either discrete or continuous.

6. BINOMIAL DISTRIBUTION POISSON DISTRIBUTION DISCRETE DISTRIBUTIONS NORMAL DISTRIBUTION CONTINUOS DISTRIBUTIONS

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. Powerpoint Templates Page 12 Exercise In Kuala Lumpur, 30% of workers take public transportation daily. In a sample of 10 workers, I.What is the probability that exactly three workers take public transportation daily? II.What is the probability that at least three workers take public transportation daily? III.Calculate the standard deviation of this distribution.

14. 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

15.  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

16. E XAMPLE 3.2 Given that, find

17. SOLUTIONS

18. 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.

19. SOLUTIONS

20. Powerpoint Templates Page 19 Exercise 1 Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways I.Find the probability of receiving three calls in a 5-minutes interval time. II.Find the probability of receiving more than two calls in 15 minutes.

21. Powerpoint Templates Page 20 Exercise 2 An average of 15 aircraft accidents occurs each year. Find I.The mean, variance and standard deviation of aircraft accident per month. II.The probability of no accident during a months.

22. IMPORTANT!!!! exactly two= 2

23. More than two/ Exceed two Two or more/ At least two/ Two or more

24. less than two/ Fewer than two At most two/ Two or fewer/ Not more than two