11_Continuous Probability Distributions Part 2-1

# Used to model quantities which are equally likely to

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used to model quantities which are “equally likely to take any value” in a given range. Examples: Random number generators are usually uniform Can use to estimate the impact of variability when only range information is available (e.g. min and max capacity in a new plant) Properties and Assumptions: 1. Also symmetric about the mean; mean = median 2. All values in range are equally likely, so no mode (alternately, every value in the range is a mode)

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14 All values equally likely in the range [a,b] Probability Dens ity Function (PDF) Cumulative Dis tribution Function (CDF) Mean Variance Stdev PDF CDF x Uniform distribution formula: X~ Uniform (a,b) a<b = - otherwise 0 ) ( 1 b x a if x f a b 2 ] [ b a X E + = = μ - - < = b x if b x a if a b a x a x if x F 1 ) ( ) ( 0 ) ( 12 ) ( 2 2 a b - = σ 12 ) ( 2 a b - = σ
15 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45. 25 45 ? x f(x)

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16 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45. 25 45 1/20 x f(x)
17 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45. 1. What is the probability that the demand is less than 28? 25 45 1/20 x f(x)

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18 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45. 1. What is the probability that the demand is less than 28? 25 45 1/20 x f(x) 28 What is the shaded area?
19 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45. 1. What is the probability that the demand is less than 28? P(X < 28) = (28 - 25)/(45 - 25) = 3 / 20 = 0.15 25 45 1/20 x f(x) 28 What is the shaded area?

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20 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45. 1. What is the probability that the demand is less than 28? 2. What is the probability that the demand is more than 39? P(X < 28) = (28 - 25)/(45 - 25) = 3 / 20 = 0.15 P(X > 39) = 1- P(X ≤ 39) =1 – (39 - 25)/(45 - 25) = 1 – 14/20 = 6/20 = 0.3 25 45 1/20 x f(x) What is the shaded area? 39
21 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45. 3. What is the probability that the demand is between 31 and 42? 25 45 1/20 x f(x) 31 42 What is the shaded area?

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22 Uniform distribution example: iPad mini Daily demand for iPad mini at the Apple store on Walnut Street is uniformly distributed between 25 and 45.
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