Statistics 312 Uebersax 11 Poisson Processes ctd Probability Density Functions 1 Poisson Distribution The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known average rate and independently of each other It can also be used for the number of events in other specified intervals such as distance area or volume Khan Academy videos on Poisson Processes Part 1 http www youtube com watch v 3z M6sbGIZ0 Part 2 http www youtube com watch v Jkr4FSrNEVY Optional video http www youtube com watch v 8x3pnyYCBto Sim on Poisson 1781 1840 Many or most applications of Poisson distributions involve studying the number of events that occur in a fixed time period However the concept generalizes to other applications such as the frequency of defects in a region of some material Generically it applies to studying how many events occur in an area of opportunity of fixed size Examples of Poisson Processes No of phone calls arriving at a call center within a minute No of photons arriving at a telescope No of mutations on a strand of DNA per unit length No of customers arriving at an airline ticket counter No of cars arriving at a traffic light No of viruses in 1 ml of blood Assumptions of a Poisson Process 1 Arrivals events are temporally or spatially distinct They occur one at a time not simultaneously or in different subareas of observation 1 Statistics 312 Uebersax 11 Poisson Processes ctd Probability Density Functions 2 The probability of an arrival or event occurrence is constant 3 The probability of each arrival or event is independent of all other arrivals events The Poisson probability distribution is characterized by only one parameter the average number of events per area of opportunity P X x e x X 0 1 2 x Example Suppose the number of flaws in a 100 foot roll of paper is a Poisson random variable with 10 Then the probability that there are eight flaws in a 100 foot roll is The probability of seven flaws in a 50 foot roll is Calculating the Poisson distribution in Excel POISSON x mu cumulative x No of events mu Average arrival event rate cumul 0 noncumulative distribution cumul 1 cumulative distribution Example POISSON 8 10 0 0 112599032 Mean and the Standard Deviation of a Poisson Distribution x x Example 100 foot roll 2 Statistics 312 Uebersax 11 Poisson Processes ctd Probability Density Functions x 10 x 10 3 16 Read pp 167 171 Prob 4 40 a and c only 2 Probability Distributions and Probability Density Functions Continuous random variables are variables for which any value within a specified range can occur Discrete Probability Distribution Example Pr X x x 0 1 2 3 4 5 X variable x some specific value of X 0 1 2 possible value of x Mean and Variance of a Discrete Probability Distribution Example We could consider raw data X 0 0 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 4 5 5 X 2 32 2 2 035 Or we could express the same information as a frequency distribution or relative frequency distribution probability distribution Probability Distribution Probability Distribution P X x 0 30 0 25 Probability Frequency Distribution Relative Value Frequency Frequency x f x p x 0 2 0 09 1 5 0 23 2 6 0 27 3 4 0 18 4 3 0 14 5 2 0 09 Total 22 1 0 0 20 0 15 0 10 0 05 0 00 0 1 2 3 4 5 x In Lecture 10 we learned that we can compute the mean expected value and variance directly from a summary table like that above 3 Statistics 312 Uebersax 11 Poisson Processes ctd Probability Density Functions If we apply these formulas to the tabled data we will get the same results as with calcualting the mean and variance directly from the raw data So when we talk about finding the expected value and variance of a probability distribution figure on right it s basically the same as finding the mean and variance of the raw unaggregated variable Continuous Probability Distributions Video An Introduction to Continuous Probability Distributions http www youtube com watch v OWSOhpS00 s Recall what a discrete probability distribution looks like e g binomial distribution Imagine narrower and narrower x axis intervals converging on infinitely narrow intervals i e a probability distribution for a continuous variable 4 Statistics 312 Uebersax 11 Poisson Processes ctd Probability Density Functions However because we have infinitely many infinitely small intervals on the x axis the y axis no longer reflects probability we ll see why shortly Instead the y axis is called probability density The probability density function of x f x or pdf x supplies the probability density y for each possible value of x In general for a continuous variable the probability of x falling between a and b is b P a X b a f x dx Probability density function is abbreviated as pdf The y axis of a pdf is rescaled so that the total area under the curve e g from inf to inf is 1 0 This means the y axis height is somewhat arbitrary It is more or less a scaling factor needed to assure that the total area under the curve is 1 0 This means that the height of pdf x is not the probability of x occurring It is the probability density of x However the probability density is proportional to probability The ratio of pdf x1 to pdf x2 is the same as the ratio of Pr x1 to Pr x2 5 Statistics 312 Uebersax 11 Poisson Processes ctd Probability Density Functions The cumulative distribution function gives the area probability of being less than a given value of x This function is denoted as F x or cdf x b P X b F b f x dx The mean or expected value and variance for a continuous variable E X xf x dx 2 x f x dx 2 The uniform or rectangular distribution is a pdf where all values of x are equally likely 6 Statistics 312 Uebersax 11 Poisson Processes ctd Probability Density Functions We denote the lower and upper x axis bounds as a and b respectively Note a and b here define an entire range of possible values above they delimited only a specific subrange The expected value and variance of a uniform is given by The probability that x falls into a range defined by xlow and xhigh is Pr xlow x xhigh xhigh x xlow b a Videos Playlist Continuous Probability Distributions http www youtube com playlist list PLvxOuBpazmsPDZGwqhhjE3KkLWnTD34R0 Read pp 180 183 7