Discrete Probability Distributions

1. Discrete Probability Distributions

2. Random Variable • Random experiment is an experiment with random outcome. • Random variable is a variable related to a random event

3. Discrete - Continuous • Random variable is discrete if it can take no more than countable number of values • Random variable is continuous, if it can take any value in an interval

4. Discrete Random Variables • The number of throws of a coin needed before a head first appears • The number of dots when rolling a dice • The number of defective items in a sample of 20 items • The number of customers arriving at a check-out counter in an hour • The number of people in favor of nuclear power in a survey

5. Continuous Random Variables • The yearly income for a family • The amount of oil imported into Finland in a particular month • The time that elapses between the installation of a new component and its failure • The percentage of impurity in a batch of chemicals

6. Discrete probability distribution • Discrete random variable values and their probabilities.

7. Fortune wheel If the probability to win when rolling a fortune wheel is 15% then the probability distribution for the number of wins in 5 rolls is:

8. Two Dice

9. x F(x) 2 1/36 3 3/36 4 6/36 5 10/36 6 15/36 7 21/36 8 26/36 9 30/36 10 33/36 11 35/36 12 36/36 Cumulative Distribution Cumulative distribution function F(x) equals the probability to get at most x. When playing two dice the sum of outcomes lies between 2-12. Using cumulative distribution we can easily find probabilities for different events: P(X<7) = 15/36  0,42 P(X>9) = 1 – 30/36 = 6/36  0,17 P(4<X<9) = 26/36 – 6/36 = 20/36  0,56

10. Expected Value • Expected value is just like the mean in empirical distributions Examples: • When playing a dice the expected value equals 3,5 • Insurance company is interested in the expected value of indemnities • Investor is interested in the expected value of portfolio’s revenue

11. Expected value calculation • The expected value for a discrete random variable is obtained by multiplying each possible outcome by its probability and then sum these products

12. Expected value example 1 • Annual costs of an investment are estimated to be 100 000 per year for next 10 years. • Under boom estimated revenue is 180 000 per year and under recession 110 000 per year. • Probability of boom is 0,40 and probability of recession is 0,60. Estimate the profitability of the investment.

13. Expected Value example 2 • Assume a lottery with 1000 lottery and 31 winningtickets. One ticket wins 500, ten tickets win 300 and 20 tickets win 100. • Define the ticket price so that the expected value of the win is 55% of the ticket price.

14. Expected value example 3 According to manufacturer’s statistics the car model needs repairs under warranty as follows: • No repairs for 50% of cars • On the average 100 euros repairs for 20% of cars • On the average 200 euros repairs for 25% of cars • On the average 500 euros repairs for the rest of the cars How much should the warranty increase the price of the car?

15. Expected Value example 4 • An arranger of a sports event wants to take a rain insurance. The insurance price is defined using the probabilities of rain and the amounts of possible indemnities. • Define the price so that it is 40% higher than the expected value of indemnity.

16. Binomial Distribution Bin(n,p) • The experiment consists of a sequence of n identical trials • All possible outcomes can be classified into two categories, usually called success and failure • The probability of an success, p, is constant from trial to trial • The outcome of any trial is independent of the outcome of any other trial Binomial experiments satisfy the following:

17. Binomial Distribution Random Variables • The number of heads when tossing a coin for 50 times • The number of reds when spinning the roulette wheel for 15 times • The number of defective items in a sample of 20 items from a large shipment • The number of people in favour of nuclear power in a survey

18. Binomial distribution and Excel You can use Excel to find probabilities related to binomial distribution random variables (the number of successes x in the n trials: • Probability =BINOMDIST(x;n;p;0) • Cumulative probability =BINOMDIST(x;n;p;1)

19. Poisson distribution Poisson experiments satisfy the following • The probability of occurrence of an event is the same for any two intervals of equal length • The occurrence or non-occurrence of the event in any interval is independent of the occurrence or non-occurrence in any other interval • The probability that two or more events will occur in an interval approaches zero as the interval becomes smaller

20. Poisson Distribution Random Variables • The number of failures in a large computer system during a given day • The number of ships arriving at a loading facility during a six-hour loading period • The number of delivery trucks to arrive at a central warehouse in an hour • The number of dents, scratches, or other defects in a large roll of sheet metal • The number of accidents at a crossroads during one year

21. Poisson and Excel You can use Excel to find probabilities related to Poisson distribution random variables (the number of occurrences x in an interval): • Probability =POISSON(x;;0) • Cumulative probability =POISSON(x; ;1)  = the average number of occurrences in an interval

22. Continuous Probability Distributions

23. Normal Distribution Many continuous variables are approximately normally distributed • Measurement errors • Physical and mental properties of people • Properties of manufactured products • Daily revenues of investments

24. Normal Distribution Normal distribution is defined by density function area under density function equals 1, area represents probability expected value

25. Cumulative Probability Function • Cumulative function for x = area to the left of x = probability to get at most x: x

26. Standardized Distribution N(0,1) • Cumulative function values have been tabulated (in most statistics textbooks) for normal distribution with expected value 0 and standard deviation 1 • This distribution is called standardized distribution and is denoted N(0,1). 0

27. Standardized Distribution and Excel • Cumulative probability =NORMSDIST(z) • Random variable value z =NORMSINV(probability)

28. Standardizing You can standardize any normal distribution N(,) variable to a standardized distribution N(0,1) variable SAME AREA! SAME PROB.! x  z 0

29. Normal Distribution N(,) and Excel • Excel: =NORMDIST(x;;;1) • Excel: =NORMINV(cumulative probability;;)