PROBABILITY AND STATISTICS

WEEK 7

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Normal Distributions

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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

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Normal Distributions

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StandardizationStandart Normal Distribution

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

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Standard Normal DistributionProperties:• 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

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0 145. z

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

P z( . ) .0 145 0 4265

...

...

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

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Example

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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;

a)Between 43 and 47 daysb)Less then 42 days.c)More then 49 days.

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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.

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Approximation to Normal Distribution

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Normal Approximation to the Binomial Distributions

n=20 and p=0.6

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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

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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?

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Normal Approximation to the Poisson Distributions

The normal approximation is applicable to a Poisson if λ > 5Accordingly, 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

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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?

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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.

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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)

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Examples

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Example

For a product daily avarege sales are 36 and standard deviation is 9. (The sales have normal distribution)

a)Whats the probability of the sales will be less then 12 for a day?

b)The probability of non carrying cost (stoksuzluk maliyeti) to be maximum 10%, How many products should be stocked?

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Example

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Example

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Example

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