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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