distributions.ppt - © Aki Taanila1 Discrete Probability Distributions

distributions.ppt - © Aki Taanila2 Random Variable Random experiment is an experiment with random outcome. Random variable is a variable related to a random event

distributions.ppt - © Aki Taanila3 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

distributions.ppt - © Aki Taanila4 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

distributions.ppt - © Aki Taanila5 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

distributions.ppt - © Aki Taanila6 Discrete probability distribution Discrete random variable values and their probabilities.

distributions.ppt - © Aki Taanila7 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: number of winsprobability 044,3705% 139, ,8178% 32,4384% 40,2152% 50,0076%

distributions.ppt - © Aki Taanila8 Two Dice

distributions.ppt - © Aki Taanila9 Cumulative Distribution xF(x) 21/36 33/36 46/36 510/36 615/36 721/36 826/36 930/ / / /36 Cumulative distribution function F(x) equals the probability to get at most x. When playing two dice the sum of outcomes lies between 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

distributions.ppt - © Aki Taanila10 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

distributions.ppt - © Aki Taanila11 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

distributions.ppt - © Aki Taanila12 Expected value example 1 Annual costs of an investment are estimated to be per year for next 10 years. Under boom estimated revenue is per year and under recession per year. Probability of boom is 0,40 and probability of recession is 0,60. Estimate the profitability of the investment.

distributions.ppt - © Aki Taanila13 Expected Value example 2 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.

distributions.ppt - © Aki Taanila14 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?

distributions.ppt - © Aki Taanila15 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. Rain (mm)ProbabilityIndemnity 0-250% % % %6000

distributions.ppt - © Aki Taanila16 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:

distributions.ppt - © Aki Taanila17 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

distributions.ppt - © Aki Taanila18 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)

distributions.ppt - © Aki Taanila19 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

distributions.ppt - © Aki Taanila20 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

distributions.ppt - © Aki Taanila21 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

distributions.ppt - © Aki Taanila22 Continuous Probability Distributions

distributions.ppt - © Aki Taanila23 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

distributions.ppt - © Aki Taanila24 Normal Distribution Normal distribution is defined by density function expected value area under density function equals 1, area represents probability

distributions.ppt - © Aki Taanila25 Cumulative Probability Function Cumulative function for x = area to the left of x = probability to get at most x: x

distributions.ppt - © Aki Taanila26 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

distributions.ppt - © Aki Taanila27 Standardized Distribution and Excel Cumulative probability =NORMSDIST(z) Random variable value z =NORMSINV(probability)

distributions.ppt - © Aki Taanila28 Standardizing You can standardize any normal distribution N( ,  ) variable to a variable of the standardized distribution N(0,1)  0xz SAME AREA! SAME PROB.!

distributions.ppt - © Aki Taanila29 Normal Distribution N( ,  ) and Excel Excel: =NORMDIST(x;  ;  ;1) Excel: =NORMINV(cumulative probability;  ;  )

distributions.ppt - © Aki Taanila30 Binomial - Normal If the number of trials is big then you can approximate binomial distribution with normal distribution: When the number of trials increases then the approximation becomes better.