The values of an exponential random variable cannot

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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 = 0.25 Ø 6.4 – UNIFORM PROBABILITY DISTRIBUTIONS o Continuous 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| o Continuous 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 variable o An 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 occurring o Probability 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

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