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

Exponential Dist. Poisson Dist. IME 312

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 Relation between Exponential distribution ↔ Poisson distribution X i : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 Y i : 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

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

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

Application of Gamma Distribution K = shape parameter >0 = number of Y i added = scale parameter >0 = rate if X ~ Expo ( ) and Y = X 1 + X 2 + ……… + X k 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

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 Relation between Exponential distribution ↔ Gamma distribution X i : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 Y i : Continuous random variable, time taken for 3 customers to arrive, has Gamma distribution with shape parameter k = 3 and scale=4 IME 312 Y 1 =1 Y 2 =7/8 Y 4 =3/4 Y 3 =1/2

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

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

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

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

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 means will have a normal distribution with a mean of and a standard deviation of. IME 312

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