EQT 272 PROBABILITY AND STATISTICS ROHANABINTIABDULHAMID INSTITUT E

EQT 272 PROBABILITY AND STATISTICS ROHANABINTIABDULHAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS Free Powerpoint Templates

CHAPTER 3 PROBABILITY DISTRIBUTION

PROBABILITY DISTRIBUTION • 3. 1 Introduction Binomial distributi on • 3. 2 • 3. 3 Poisson distributio n Normal distributio n • 3. 4

3. 1 INTRODUCTION ü ü 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.

• BINOMIAL DISTRIBUTION DISCRETE DISTRIBUTIONS • POISSON DISTRIBUTION CONTINUOS • NORMAL DISTRIBUTIONS

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.

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

Definition 3. 3 : The Mean and Variance of X If X ~ B(n, p), then Mean Variance where q n is the total number of trials, q p is the probability of success and q q is the probability of failure. Standard deviation

EXAMPLE 3. 1

SOLUTIONS

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. Powerpoint Templates Page 12

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 q

q q 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

EXAMPLE 3. 2 Given that , find

SOLUTIONS

EXAMPLE 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.

SOLUTIONS

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. Powerpoint Templates Page 19

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. Powerpoint Templates Page 20

IMPORTANT!!!! exactly two =2

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

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