 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

Continuous Probability Distributions Binomial Poisson Probability Distributions Discrete Probability Distributions Normal

Example 1.24  Check whether the distribution is a probability distribution.  Solution  # so the distribution is not a probability distribution. X01234 P(X=x)

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: 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

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.

Example 1.25

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 the following guidelines can be used:

Example 1.26 In a Binomial Distribution, n =12 and p = 0.3. Find the following probabilities. a) b) c) d) e)

Example 1.26  In a Binomial Distribution, n =12 and p = 0.3. Find the following probabilities. a) b) c) d) e)

Exercise 1.5: 1. Let X be a Binomial random variable with parameters n = 20, p = 0.4. By using cumulative binomial distribution table, find : a) b) c) d) e) f) 2. Use cumulative binomial table for n = 5 and p = 0.6 to find the probabilities below: a) ans : b) ans :

Example 1.27: In a recent year, 10% of the luxury car sold were black. If 10 cars are randomly selected, find the following probabilities : a) At least five cars are black b) At most three cars are black c) Exactly four cars are black d) Less than one are not black

Exercises 1.6 : 1. Consider a binomial random variable with n = 8 and p = 0.3. Fill in table below: 2. Fill in the blanks of table below: k ProblemWrite the probability Rewrite the probability (if needed) Find the probabillity Less than three More than three At most three Fewer than three Between 3 and 5 Exactly three At least five

Exercises 1.7 : 1. Suppose you will be attending 6 hockey games. If each game will go to overtime with probability 0.10, find the probability that a) At least 1 of the games will go to overtime. b) 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 a) Exactly two b) 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 A random variable X having a Poisson distribution can also be written as

Example 1.28 Consider a Poisson random variable with. Calculate the following probabilities : a) Write the distribution of Poisson b) c) d)

Exercises 1.8: 1. Consider a Poisson random variable with. Fill in table below: 2. Fill in the blanks of table below: k ProblemWrite the probability Rewrite the probability (if needed) Find the probabillity Less than three More than three At most three Fewer than three Between 3 and 5 Exactly three At least five

Example 1.29 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 a most three accidents on this section of highway during a 2-week period.

Solution: i) ii)

Exercise 1.9: 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 10 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, and either or Example 1.30: Given that, find : a) b)

Exercise 1.10: 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: ) ii) what is the probability that at least three workers take public transportation daily? (ans: )

3. Let Using Poisson distribution table, find i) (ans: , ) ii) (ans: , ) iii) (ans: ) 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 Compute the probability that: i) At least 5 watches will need to warranty work. (ans: ) ii) At most 3 watches will need warranty work. (ans: ) iii) Less than 7 watches will need warranty work. (ans: )