1 / 13

Exponential Distribution .

Exponential Distribution. = mean interval between consequent events = rate = mean number of counts in the unit interval > 0 X = distance between events >0. f(x). F(x). Memoryless property. Exponential and Poisson relationship. Unit Matching between x and !. IME 312.

liza
Download Presentation

Exponential Distribution .

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Exponential Distribution. = mean interval between consequent events = rate = mean number of counts in the unit interval > 0 X = distance between events >0 f(x) F(x) Memoryless property Exponential and Poisson relationship Unit Matching between x and ! IME 312

  2. Exponential Dist. Poisson Dist. IME 312

  3. Relation betweenExponential distribution ↔ Poisson distribution Xi : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8 0 1:00 2:00 3:00 Y1=3 Y2=4 Y3=5 Yi : Discrete random variable, number of arrivals per unit of time, has Poisson distribution with mean = 4. (rate=4) Y ~ Poisson (4) IME 301 and 312

  4. Continuous Uniform Distribution f(x) a b F(x) a b IME 312

  5. Gamma Distribution K = shape parameter >0 = scale parameter >0 For Gamma Function, you can use: and if K is integer (k’) then: IME 312

  6. Application of Gamma Distribution K = shape parameter >0 = number of Yi added = scale parameter >0 = rate if X ~ Expo ( ) and Y = X1 + X2 + ……… + Xk then Y ~ Gamma (k, ) i.e.: Y is the time taken for K events to occur and X is the time between two consecutive events to occur IME 312

  7. Relation betweenExponential distribution ↔ Gamma distribution Xi : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8 0 1:00 2:00 3:00 Y1=1 Y2=7/8 Y3=1/2 Y4=3/4 Yi : Continuous random variable, time taken for 3 customers to arrive, has Gamma distribution with shape parameter k = 3 and scale=4 IME 312

  8. Weibull Distribution a = shape parameter >0 = scale parameter >0 for for IME 312

  9. Normal Distribution Standard Normal Use the table in the Appendix IME 312

  10. Normal Approximation to the Binomial Use Normal for Binominal if n is large X~Binomial (n, p) Refer to page 262 IME 312

  11. Central Limit Theorem : random sample from a population with and : sample mean Then has standard normal distribution N(0, 1) as commonly IME 312

  12. What does Central Limit Theorem mean? Consider any distribution (uniform, exponential, normal, or …). Assume that the distribution has a mean of and a standard deviation of . Pick up a sample of size “n” from this distribution. Assume the values of variables are: Calculate the mean of this sample . Repeat this process and find many sample means. Then our sample meanswill have a normal distribution with a mean of and a standard deviation of . IME 312

  13. = degrees of freedom = probability Distribution Definition Notation Chi-Square t dist. F dist. Where: IME 312

More Related