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The values of an exponential random variable cannot be negative (can’t arrive -2 minute after previous customer)-Normal random variables can be negative●Calculating Exponential Probabilities○Exponential cumulative distribution function■P(x £ a) = 1 – e^(-a • mean)-e = 2.7818-mean = mean number of occurrences over the interval-a = any number of interest■Finding the Mean-Always a countable rate-µ = measurable interval (such as 4 minutes b/w customer arrivals)-1/u = mean-¼ = 0.25 (in example below)○Average Time (mean) b/w customer arrivals is 4 minutes – what is the probability
that the next customer will arrive w/in 2 minutes?■Looking for probability of LESS THAN or EQUAL TO 2■P(x £ a) = 1 – e^(-a • mean)-1 – e^(-2 • 0.25) = 0.3935-à 39% chance that next customer will arrive w/in next 2 minutes○Probability that next customer will arrive between 4 – 8 minutes from now■P(4 £ a £ 8) = P(x £ 8) – P(x £ 4)■P(x £ 8) = 1 – e^(-8 • 0.25) = 0.8647■P(x £ 4) = 1 – e^(-4 • 0.25) = 0.6321■DIFF: (0.8647 – 0.6321) = 0.2326 or 23%○Formula for Variance of Exponential Distribution■σ² = µ = 1/mean■square root of the µ (mean)■in example, µ = 4 à σ = Ö4 = 2.0-standard deviation is 2.0●Calculating Exponential Probabilities Using Excel○The Function■= EXPONDIST(x, lambda, cumulative)-Cumulative = FALSE – if want the probability density function-Cumulative = TRUE – if want the cumulative probability○Another Note■Terms µ and l (average rate of arrivals over a specific interval) must be based on the same units■Lambda (in function) = 1/µ-In example = 0.25Ø6.4 – UNIFORM PROBABILITY DISTRIBUTIONSoContinuous uniform probability distribution– the probability of any interval in the distribution is equal to any other interval with the same width§The graph that looks like a pair of pants |ˉˉˉ|oContinuous uniform probability distribution function–§f(x) = 1 / (b – a) IF (a £ x £ b)-if the IF is not true, f(x) = 0-a = minimum continuous random variable-b = maximum continuous random variableoAn Example:§The Story:-time it takes to play a round of golf at the Golf Club has minimum time of 4 hours (240 minutes) and max time of 4 hours and 50 minutes (290 minutes)
§The Facts-Any 1-minute interval w/in this range of time (240 – 290 minutes) has the same probability of occurringoProbability that golf round will take > 4 hours & 12 minutes (252 minutes)■Finding the variables-Min (a) = 240-Max (b) = 290-X = 252 (definitely falls w/in A and B)■F(x) = 1 / (290 – 240) = 0.02-0.2 will be height (on y-axis) of the uniform probability function graph-Refers to probability of a one-unit interval w/in the distribution occurring-THUS, probability w/in the max and min (240 and 290) will always equal 2%○Uniform Cumulative Distribution Function– probability that a continuous uniform random variable will fall b/w 2 particular values-Where the two values being considered are x1 and x2■P(x1 £ x £ x2) = (x2 – x1) ¸ (b – a)-Calculates the area of the distribution b/w values x1 and x2○Probability the next round of golf will take LESS THAN 252 minutes