# Occurrences in a fixed time period if the rate that

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occurrences in a fixed time period if the rate that events occur is constant. A random variable that follows a Poisson Distribution is called a Poisson Process . If occurrences follow a Poisson Process with mean = , then the waiting time for the next occurrence has Exponential distribution with mean = 1/ . Example - accidents at an oil refinery 68 Accidents occur at an oil refinery at a constant rate of 3 per month. This is an example of a Poisson Process. The random variable Y = the number of accidents in the next month would follow a Poisson Distribution with    occurrences per month The Random Variable X = the waiting time until the next refinery accident would follow an Exponential distribution with    months. Find the probability of waiting less than 2 months for the next oil refinery accident. 2 1/3 6 2 1 1 0.9975 P X e e
P a g e | 112 Find the 90 th percentile of waiting times for a refinery accident. 1 95 3 ln 1 .90 0.768 x   months 6.5 Uniform Distribution A uniform distribution is a continuous random variable in which all values between a minimum value and a maximum value have the same probability. The two parameters that define the Uniform Distribution are: a= minimum b = maximum The probability density function is the constant function f(x) = 1/(b-a), which creates a rectangular shape. Example - loose leaf tea A tea lover enjoys Tie Guan Yin loose leaf tea and drinks it frequently. To save money, when the supply gets to 50 grams he will purchase this popular Chinese tea in a 1000 gram package. The amount of tea currently in stock follows a uniform random variable. X = the amount of tea currently in stock a = minimum = 50 grams b = maximum = 1050 grams f(x) = 1/(1050 - 50) = 0.001
P a g e | 113 The expected value, population variance and standard deviation are calculated using the formulas: 2 2 2 2 12 12 b a b a a b For the loose leaf tea problem: 50 1050 550 2 g 2 2 1050 50 83,333 12 83333 289 g Probability problems can be easily solved by finding the area of rectangles. Find the probability that there are at least 700 grams of Tie Guan Yin tea in stock. ( 700) (1050 700)(0.001) 0.35 P X width height The pth percentile of the Uniform Distribution is calculated by using linear interpolation: p x a p b a Find the 80th percentile of Tie Guan Yin in stock: 80 50 0.80 1050 50 850 x grams
P a g e | 114 The important features of the Uniform Distribution are summarized here: Example - waiting for a train The Sounder commuter train 69 from Lakeview to Seattle, Washington arrives at Tacoma station every 20 minutes during the morning rush hour. Assume that this train is running on time. Let X = the waiting time for the next train to arrive. X will follow a Uniform Distribution with the minimum waiting time of 0 minutes (you just catch the train) and a maximum waiting time of 20 minutes (you just miss the train).