1. PROBABILITY AND STATISTICS
WEEK 8
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2. Continuous Random Variables and Probability Distributions
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3. Example
Suppose that the probability density function of X is; Determine P(X < 2) , P(2 ≤ X < 4) , and P(X≥4)
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4. Cumulative Distribution Functions
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5. Example
Determine the cumulative distribution function of X. (for previous question)
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6. Mean and Variance of a Continuous Random Variable
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7. Example
Determine the mean, variance, and standard deviation of X. (for previous question)
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8. Continuous Uniform Distribution
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9. The Exponential Distributions
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10. 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?
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11. Normal Distributions
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12. 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
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13. Normal Distributions
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14. Normal Distributions
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15. 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.
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16. 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 = and z = 3.00
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17. Standardization Standart Normal Distributions
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18. 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 .
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19. 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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20. Example
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21. 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.
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22. 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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23. Approximation to Normal Distribution
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24. Normal Approximation to the Binomial Distributions
n=20 and p=0.6
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25. 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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26. 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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27. 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
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28. 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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29. 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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30. 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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31. Examples
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32. 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?
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33. Example
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34. Example
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35. Example
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