1.  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

2. Continuous Probability Distributions Binomial Poisson Probability Distributions Discrete Probability Distributions Normal

3. 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 Binomial Distribution

4. 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.

5. Example

6. Solutions:

7. 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:

8. Example:

10.  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. The Poisson Distribution

11. 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

12. Given that, find Example :

14. Poisson Approximation of Binomial Probabilities Example:

15.  Numerous continuous variables have distribution closely resemble the normal distribution.  The normal distribution can be used to approximate various discrete prob. dist. The Normal Distribution

16.  ‘Bell Shaped’  Symmetrical  Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median = Mode X f(X) μ σ The Normal Distribution

17. By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

18. The Standard Normal Distribution  Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (Z)  Need to transform X units into Z units using  The standardized normal distribution (Z) has a mean of 0, and a standard deviation of 1,  Z is denoted by

19. Patterns for Finding Areas under the Standard Normal Curve

20. Example

21. Determine the probability or area for the portions of the Normal distribution described. Exercises

22. solutions

23. Example

24. Exercises

25. solutions

26. Suppose X is a normal distribution N(25,25). Find Example solutions

27. Exercises

28. Example

29.  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 Normal Approximation of the Binomial Distribution

30. 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: Continuous Correction Factor

31. Example 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. Solutions:

32. Suppose that 5% of the population over 70 years old has disease A. Suppose a random sample of 9600 people over 70 is taken. What is the probability that fewer than 500 of them have disease A? Exercises

33.  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 Normal Approximation of the Poisson Distribution

34. 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? Example Solutions: