Discrete Probability Distributions 1 Random Variable l l

Discrete Probability Distributions 1

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

Discrete - Continuous l l 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 3

Discrete Random Variables l l l 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 4

Continuous Random Variables l l 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 5

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

Fortune wheel If the probability to win when number of wins 0 rolling a fortune wheel is 15% then the probability 1 distribution for the number 2 of wins in 5 rolls is: 3 probability 44, 3705% 39, 1505 13, 8178% 2, 4384% 4 0, 2152% 5 0, 0076% 7

Two Dice 8

Cumulative Distribution Cumulative distribution function F(x) equals the probability to get at most x. x 2 3 4 5 6 7 8 9 10 11 12 F(x) 1/36 3/36 6/36 10/36 15/36 21/36 26/36 30/36 33/36 35/36 36/36 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 9

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

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

Expected value example 1 l l l 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. 12

Expected Value example 2 l l Assume a lottery with 1000 lottery and 31 winning tickets. 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. 13

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

Expected Value example 4 l l 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. Rain (mm) Probability Indemnity 0 -2 50% - 3 -5 30% 500 6 -10 18% 2000 11 - 2% 6000 15

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

Binomial Distribution Random Variables l l 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 17

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: l. Probability =BINOMDIST(x; n; p; 0) l. Cumulative probability =BINOMDIST(x; n; p; 1) 18

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

Poisson Distribution Random Variables l l l 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 20

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

Continuous Probability Distributions 22

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

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

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

Standardized Distribution N(0, 1) l l 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 26

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

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

Normal Distribution N( , ) and Excel l l Excel: =NORMDIST(x; ; ; 1) Excel: =NORMINV(cumulative probability; ; ) 29