Probability Distributions

1. 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.

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

3. 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: • No. of getting a head in tossing a coin 10 times. • No. of getting a six in tossing 7 dice. • A firm bidding for contracts will either get a contract or not

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 Xif 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:

9. Example:

10. The 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: • No. of cars passing a toll booth in one hour. • No. defects in a square meter of fabric • No. of network error experienced in a day.

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. Example : Given that , find

13. Example :

14. Poisson Approximation of Binomial Probabilities Example:

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

16. The Normal Distribution • ‘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   f(X) σ X μ Mean = Median = Mode

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

18. The Standard Normal Distribution • Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normaldistribution (Z) • Need to transform X units into Zunits 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. Exercises Determine the probability or area for the portions of the Normal distribution described.

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23. Example

24. Exercises

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

27. Exercises

28. Example

29. Normal Approximation of the Binomial Distribution • 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

30. Continuous Correction Factor 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:

31. Solutions: 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.

32. Exercises 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?

33. Normal Approximation of the Poisson Distribution • 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

34. Example 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? Solutions: