Distributions

1. Distributions Onur DOĞAN

2. SpecialDistributions Onur DOĞAN

3. ContiniousUniformDistributions • asdaf.

4. Example Suppose that a random number generator produces real numbers that areuniformly distributed between 0 and 100. • Determine the probability density function of a random number (X) generated. • Find the probability that a random number (X) generated isbetween 10 and 90. • Calculate the mean and variance ofX.

5. The Exponantial Distributions • ljhlj

6. Example The number of customers who come to a donut store follows a Poissonprocess with a mean of 5 customers every 10 minutes. • Determine theprobability density function of the time (X; unit: min.) until the next customerarrives. • Find the probability that there are no customers for at least 2minutes by using the corresponding exponential and Poisson distributions. • Howmuch time passes, until the next customer arrival • Findthevariance?

7. Normal Distributions • .

8. Normal Distributions

9. StandardNormalRandomVariable The standard normal random variable (denoted as Z) is a normal random variablewith mean µ= 0 and variance Var(X) = 1.

10. Standardization

11. Readingthe Z Table • P(0 ≤ Z ≤ 1,24) = • P(-1,5 ≤ Z ≤ 0) = • P(Z > 0,35)= • P(Z ≤ 2,15)= • P(0,73 ≤ Z ≤ 1,64)= • P(-0,5 ≤ Z ≤ 0,75) = • Find a value of Z, say, z , such that P(Z ≤ z)=0,99

12. Example

13. Example • A debitor pays back his debt with the avarage 45 days and variance is 100 days. Find the probability of a person’s paying back his debt; • Between 43 and 47 days • Less then 42 days. • More then 49 days.

14. Normal Approximation to theBinomial Distributions The binomial distribution B(n,p)approximates to the normal distribution with E(x)= np and Var(X)= np(1 - p) if np > 5 and n(l -p) > 5

15. Example Suppose that X is abinomial random variable with n = 100 andp = 0.1. Find the probability P(X≤15) based on the corresponding binomialdistribution and approximate normaldistribution. Is the normal approximation reasonable?

16. Binomial Form

17. Normal Approximation to thePoissonDistributions The normal approximation is applicable to a Poissonif λ > 5 Accordingly, when normal approximation is applicable, the probability of aPoisson random variable X with µ=λand Var(X)= λ can be determined by using thestandard normal random variable

18. Example Suppose that X has aPoisson distribution with λ= 10. Find the probability P(X≤15) based on thecorresponding Poisson distribution and approximate normal distribution. Is thenormal approximation reasonable?

19. Poisson Form

20. Normal Approximation to theHypergeometricDistributions Recall that the binomial approximation is applicable to a hypergeometricif the sample size n is relatively small to the population size N, i.e.,to n/N < 0.1. Consequently, thenormal approximation can be applied to the hypergeometric distribution withp =K/N (K: number of successes in N) if n/N < 0.1, np > 5. and n(1 - p) > 5.

21. Example Suppose that X hasa hypergeometric distribution with N = 1,000, K = 100, and n = 100. Find theprobability P(X≤15) based on the corresponding hypergeometric distribution and approximate normal distribution. Is the normal approximation reasonable?

22. Hypergeometric Form

23. Examples

24. Example

25. Solution

26. Example

28. Example

29. Solution

30. Example

31. Solution

32. Example

33. Solution

34. Example • For a product daily avarege sales are 36 and standard deviation is 9. (The sales have normal distribution) • Whats the probability of the sales will be less then 12 for a day? • The probability of non carrying cost (stoksuzluk maliyeti) to be maximum 10%, How many products should be stocked?