Chapter 3 Probability Distribution

Chapter 3 Probability Distribution

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.

Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Poisson

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: Binomial distribution is written as X ~ B(n,p) 1. No. of getting a head in tossing a coin 10 times. 2. A firm bidding for contracts will either get a contract or not

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 ( ) n x n P X x C p q x ; x 0,1,2... n The Mean and Variance of X if X ~ B(n,p) are x Mean : E( X ) np Variance : V( X ) np(1 p) npq 2 Std Deviation : npq where n is the total number of trials, p is the probability of success and q is the probability of failure.

Given that X ~ B 12,0.4 a) P X 2 b) P X 4 c) E X d) Var( X ) Example

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 P X k the following guidelines can be used:

Cumulative Binomial Distribution In a Binomial Distribution, n =12 and p = 0.3. Find the following probabilities. a) b) c) d) PX ( 5) PX ( 5) PX ( 9) P(5 X 9) e) P(3 X 5)

Exercise 1. Let X be a Binomial random variable with parameters n = 20, p =0.4. By using cumulative binomial distribution table, find : a) PX ( 14) b) PX ( 10) c) P(7 X 15) 2 Let X be a Binomial random variable with n = 5 and p = 0.6, find the probabilities below: a) PX ( 3) b) PX ( 3)

Example The probability that a boards purchased by a cabinet manufacturer are unusable for building cabinets is 0.10. The cabinet manufacturer bought eleven boards, what is the probability that i. Four or more of the eleven boards are unusable for building cabinets? ii. iii. At most two of the eleven boards are unusable for building cabinets? None of the eleven boards are unusable for building cabinets?

Exercise 1. Suppose you will be attending 6 hockey games. If each game will go to overtime with probability 0.10, find the probability that i. At least 1 of the games will go to overtime. ii. At most 1 of the games will go to overtime. 2. Statistics indicate that alcohol is a factor in 50 percent of fatal automobile accidents. Of the 3 fatal automobile accidents, find the probability that alcohol is a factor in i. Exactly two ii. At least 1

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: 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.

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 x e P( X x) for x 0,1,2,3,... x! A random variable X having a Poisson distribution can also be written as X P ( ) o with E( X ) and Var( X )

Example Consider a Poisson random variable with mean equal to three. Calculate the following probabilities : i. Write the distribution of Poisson ii. P(X=0) iii. P(X=1) iv. P(X >1)

Example The average number of traffic accidents on a certain section of highway is two per week. Assume that the number of accidents follows a Poisson distribution with mean is 2. i) Find the probability of no accidents on this section of highway during a 1-week period ii) Find the probability of three accidents on this section of highway during a 2-week period.

Example The number of points scored by team A in a basketball match is Poisson distributes with mean, μ. If the probability that the team does not score any point in a match is 0.09, calculate i. μ ii. the probability that the team score at least 16 points in 4 matches.

Exercise 1. The demand for car rental by AMN Travel and Tours can be modelled using Poisson distribution. It is known that on average 4 cars are being rented per day. Find the probability that is randomly choosing day, the demand of car is: a) Exactly two cars b) More than three cars 2. Overflow of flood results in the closure of a causeway. From past records, the road is closed for this reason on 8 days during a 20-year period. At a village, the villagers were concern about the closure of the causeway because the causeway provides the only access to another village nearby. a) Determine the probability that the road is closed less than 5 days in 20 years period. b) Determine the probability that the road is closed between 2 and 6 days in five years period.

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 n 30, and either np 5 or nq 5

Example 3.6: Given that X~B(1000,0.004), find : a) P(X=7) b) P(X<9)

Exercise 1. Given that X ~ B(2,0.4) Find P( X 0), P( X 2), P( X 2), P( X 1), E( X ), Var( X ). (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)

3. Let X ~ P0 (12). Using Poisson distribution table, find i) P( X 8) and P( X 8) (ans: 0.1550, 0.0655) ii) P( X 4) and P( X 4) (ans: 0.9977, 0.9924) iii) P(4 X 14) (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 3 watches will need warranty work. (ans: 0.8571) iii) Less than 7 watches will need warranty work. (ans: 0.9955)

Extra exercise 1. The probability that a patient recovers from a rare blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that i. at least 10 survive ii. from 3 to 8 survive iii. exactly 5 survive 2. In a certain industrial facility accidents occur infrequently. It is known that two accidents occur on any given day and accidents are independent of each other. i. What, is the probability that there will be an accident on one day? ii. What is the probability that there are at most three accidents?

3. In a manufacturing process where glass products arc produced, defects or bubbles occur, occasionally rendering the piece undesirable: for marketing. It is known that, on average, 1 in every 1000 of these items produced has one or more bubbles. What is the probability that a random sample of 1000 will yield fewer than 7 items possessing bubbles?