PROBABILITY AND STATISTICS

PROBABILITY AND STATISTICS WEEK 7 Onur Doğan

Normal Distributions Onur Doğan

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

Normal Distributions Onur Doğan

StandardizationStandart Normal Distribution The standard normal random variable (denoted as Z) is a normal random variablewith mean µ= 0 and variance Var(X) = 1. Onur Doğan

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

StandardizationStandart Normal Distributions Onur Doğan

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

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

Example Onur Doğan

Example • A debitorpays back his debtwiththeavarage 45 days andvariance is 100 days. Findtheprobability of a person’spaying back his debt; • Between 43 and 47 days • Lessthen 42 days. • Morethen 49 days. Onur Doğan

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

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Approximation to Normal Distribution Onur Doğan

NormalApproximation to theBinomial Distributions n=20 and p=0.6 Onur Doğan

NormalApproximation to theBinomial Distributions The binomial distribution B(n,p)approximates to the normal distributionwith; E(X)= np and Var(X)= np(1 - p) if np > 5 and n(l -p) > 5 Onur Doğan

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? Onur Doğan

NormalApproximation 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 Onur Doğan

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? Onur Doğan

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 with p =K/N (K: number of successes in N) if n/N < 0.1, np > 5. and n(1 - p) > 5. Onur Doğan

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 distributionand approximate normal distribution. Is the normal approximation reasonable? (δ=2,85) Onur Doğan

Examples Onur Doğan

Example For a productdailyavaregesalesare 36 andstandarddeviation is 9. (Thesaleshave normal distribution) • Whatstheprobability of thesaleswill be lessthen 12 for a day? • Theprobability of noncarryingcost (stoksuzluk maliyeti) to be maximum 10%, Howmanyproductsshould be stocked? Onur Doğan

Example Onur Doğan

PROBABILITY AND STATISTICS