PROBABILITY AND STATISTICS WEEK 8 Onur Doğan 2016-2017

Continuous Random Variables and Probability Distributions Onur Doğan 2016-2017

Example Suppose that the probability density function of X is; Determine P(X < 2) , P(2 ≤ X < 4) , and P(X≥4) Onur Doğan 2016-2017

Cumulative Distribution Functions Onur Doğan 2016-2017

Example Determine the cumulative distribution function of X. (for previous question) Onur Doğan 2016-2017

Mean and Variance of a Continuous Random Variable Onur Doğan 2016-2017

Example Determine the mean, variance, and standard deviation of X. (for previous question) Onur Doğan 2016-2017

Continuous Uniform Distribution Onur Doğan 2016-2017

The Exponential Distributions Onur Doğan 2016-2017

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? Onur Doğan 2016-2017

Normal Distributions Onur Doğan 2016-2017

Normal Probability Distributions The normal probability distribution is the most important distribution in all of statistics Many continuous random variables have normal or approximately normal distributions Need to learn how to describe a normal probability distribution Onur Doğan 2016-2017

Normal Distributions Onur Doğan 2016-2017

Normal Distributions Onur Doğan 2016-2017

Standardization Standart Normal Distribution The standard normal random variable (denoted as Z) is a normal random variable with mean µ= 0 and variance Var(X) = 1. Onur Doğan 2016-2017

Standard Normal Distribution Properties: The total area under the normal curve is equal to 1 The distribution is mounded and symmetric; it extends indefinitely in both directions, approaching but never touching the horizontal axis The distribution has a mean of 0 and a standard deviation of 1 The mean divides the area in half, 0.50 on each side Nearly all the area is between z = -3.00 and z = 3.00 Onur Doğan 2016-2017

Standardization Standart Normal Distributions Onur Doğan 2016-2017

Example Example: Find the area under the standard normal curve between z = 0 and z = 1.45 z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 1.4 0.4265 . Onur Doğan 2016-2017

Example (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 Onur Doğan 2016-2017

Example Onur Doğan 2016-2017

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. Onur Doğan 2016-2017

Example The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours. Find the probability that the sick-leave time of an employee in a month exceeds 130 hours. Onur Doğan 2016-2017

Approximation to Normal Distribution Onur Doğan 2016-2017

Normal Approximation to the Binomial Distributions n=20 and p=0.6 Onur Doğan 2016-2017

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 Onur Doğan 2016-2017

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? Onur Doğan 2016-2017

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 Onur Doğan 2016-2017

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? Onur Doğan 2016-2017

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. Onur Doğan 2016-2017

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? (δ=2,85) Onur Doğan 2016-2017

Examples Onur Doğan 2016-2017

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? Onur Doğan 2016-2017

Example Onur Doğan 2016-2017

Example Onur Doğan 2016-2017

Example Onur Doğan 2016-2017