#### Transcript Introduction to Probability

Onur DOĞAN Onur DOĞAN asdaf. 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. ljhlj 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? . The standard normal random variable (denoted as Z) is a normal random variable with mean µ= 0 and variance Var(X) = 1. 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 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. 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 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? 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 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? 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. 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? 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?