DISTRIBUTIONS Onur DOAN SPECIAL DISTRIBUTIONS Onur DOAN Continious

DISTRIBUTIONS Onur DOĞAN

SPECIAL DISTRIBUTIONS Onur DOĞAN

Continious Uniform Distributions � asdaf.

Example Suppose that a random number produces real numbers that are distributed between 0 and 100. generator uniformly Determine the probability density function of a random number (X) generated. � Find the probability that a random number (X) generated is between 10 and 90. � Calculate the mean and variance of X. �

The Exponantial Distributions � ljhlj

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

Normal Distributions � .

Normal Distributions

Standard Normal Random Variable The standard normal random variable (denoted as Z) is a normal random variable with mean µ= 0 and variance Var(X) = 1.

Standardization

Reading the 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

Example

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.

Normal Approximation to the Binomial 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

Example Suppose that X is a binomial random variable with n = 100 and p = 0. 1. Find the probability P(X≤ 15) based on the corresponding binomial distribution and approximate normal distribution. Is the normal approximation reasonable?

Binomial Form

Normal Approximation to the Poisson Distributions The normal approximation is applicable to a Poisson if λ > 5 Accordingly, when normal approximation is applicable, the probability of a Poisson random variable X with µ=λ and Var(X)= λ can be determined by using the standard normal random variable

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

Poisson Form

Normal Approximation to the Hypergeometric Distributions Recall that the binomial approximation is applicable to a hypergeometric if the sample size n is relatively small to the population size N, i. e. , to n/N < 0. 1. Consequently, the normal approximation can be applied to the hypergeometric distribution with p =K/N (K: number of successes in N) if n/N < 0. 1, np > 5. and n(1 - p) > 5.

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

Hypergeometric Form

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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?