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Understanding Uniform & Exponential Distributions: Examples & Solutions

Explore the concepts of Uniform and Exponential Distributions with examples and solutions. Learn about probability functions, cumulative functions, mean, variance, and practical applications. Understand Normal and Exponential Distributions in statistical analysis.

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Understanding Uniform & Exponential Distributions: Examples & Solutions

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  1. Statistical Distributions

  2. Uniform Distribution A R.V. is uniformly distributed on the interval (a,b) if it probability function Fully defined by (a,b) P(x) = 1/(b-a) for a <= x <= b = 0 otherwise

  3. Uniform Distribution Probability Function 1 1/9 1 10

  4. Probability that x is between 2 and 7.5? Probability that x = 8? 1 1/9 1 10

  5. Uniform Distribution The cumulative distribution of a uniform RV is F(x) = 0 for x < a = (x-a)/(b-a) for a <= x <= b = 1 otherwise

  6. Uniform Distribution Cumulative Function 1 1 10

  7. Uniform DistributionDiscrete vs. Continuous • Discrete RV • Number showing on a die • Continuous RV • Time of arrival • When programming, make it discrete to some number of decimal places

  8. Uniform Distribution • Mean = (a+b)/2 • Variance = (b-a)2 /12 • P (x < X < y) = F (y) – F (x) = (y-a) - (x-a) = y – x – a + a = y - x b-a b-a b – a b-a

  9. Uniform - Example A bus arrives at a bus stop every 20 minutes starting at 6:40 until 8:40. A passenger does not know the schedule but randomly arrives between 7:00 and 7:30 every morning. What is the probability the passenger waits more than 5 minutes.

  10. Uniform Solution X = RV, Uniform (0,30) -- i.e. 7:00 – 7:30 Bus: 7:00, 7:20, 7:40 Yellow Box <= 5 minute wait 1 1/30 A B C 5 10 15 20 25 30 40 P (x > 5) = A + C = 1 – B = 5/6

  11. Arithmetic Mean Given a set of measurements y1, y2, y3,… yn Mean = (y1+y2+…yn) / n

  12. Variance Variance of a set of measurements y1, y2, y3,… yn is the average of the deviations of the measurements about their mean (m). V = σ2 = (1/n) Σ (yi – m)2 i=1..n

  13. Variance Example Yi= 12, 10, 9, 8, 14, 7, 15, 6, 14, 10 m = 10.5 V= σ2 = (1/10)((12-10.5)2 + (10-10.5)2 +…. = (1/10) (1.52 + .52 + 1.52….) = (1/10) (88.5) = 8.85 Standard Deviation = σ = 2.975

  14. Normal Distribution • Has 2 parameters • Mean - μ • Variance – σ2 • Also, Standard deviation - σ

  15. Normal Dist. .3413 .1359 .0215 .0013 -3 -2 -1 1 2 3 0 Mean +- n σ

  16. Normal Distribution • Standard Normal Distribution has • Mean = 0 StdDev = 1 • Convert non-standard to standard to use the tables Z value = # of StdDev from the mean Z is value used for reading table Z = (x – m) σ

  17. Normal - Example The scores on a college entrance exam are normally distributed with a mean of 75 and a standard deviation of 10. What % of scores fall between 70 & 90? Z(70) = (70 – 75)/10 = - 0.5 Z(90) = (90 – 75)/10 = 1.5 .6915 - .5 = .1915 + .9332 - .5 = .4332 = .6247 or 62.47%

  18. Exponential Distribution A RV X is exponentially distributed with parameter > 0 if probability function Mean = 1/ Variance = 1 / 2 e = 2.71828182 P(x) = For x >= 0 = 0 Otherwise

  19. Exponential Distribution • Often used to model interarrival times when arrivals are random and those which are highly variable. • In these instances lambda is a rate • e.g. Arrivals or services per hour • Also models catastrophic component failure, e.g. light bulbs burning out

  20. Exponential Rates • Engine fails every 3000 hours • Mean: Average lifetime is 3000 hours • = 1/3000 = 0.00033333 • Arrivals are 5 every hour • Mean: Interarrival time is 12 minutes • = 1 / 5 = 0.2 • Mean = 1 /

  21. Exponential DistributionProbability Function f(x) x See handout for various graphs.

  22. Exponential DistributionCumulative Function Given Mean = 1/ Variance = 1/ 2 F(x) = P (X <=x) = 1 – e - x

  23. Exponential DistributionCumulative Function (<=) 1 F(x) x

  24. Forgetfulness Property Given: the occurrence of events conforms to an exponential distribution: The probability of an event in the next x-unit time frame is independent on the time since the last event. That is, the behavior during the next x-units of time is independent upon the behavior during the past y-units of time.

  25. Forgetfulness Example • The lifetime of an electrical component is exponentially distributed with a mean of . • What does this mean??

  26. Forgetfulness ExamplesThe following all have the same probability • Probability that a new component lasts the first 1000 hours. • Probability that a component lasts the next 1000 hours given that it has been working for 2500 hours. • Probability that a component lasts the next 1000 hours given that I have no idea how long it has been working.

  27. Solution to Example • Suppose the mean lifetime of the component is 3000 hours. • = 1/3000 • P(X >= 1000) = 1 – P(X <= 1000) 1 – (1-e -1/3* 1) = e -1/3 = .717

  28. How do we apply these? • We may be given the information that events occur according to a known distribution. • We may collect data and must determine if it conforms to a known distribution.

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