CVE SOME DISCRETE PROBABILITY DISTRIBUTIONS

Transcription

1 CVE SOME DISCRETE PROBABILITY DISTRIBUTIONS Assist. Prof. Dr. Bertuğ Akıntuğ Civil Engineering Program Middle East Technical University Northern Cyprus Campus CVE 472 Statistical Techniques in Hydrology. 1/23

2 Outline Hypergeometric Distribution Binomial Distribution Geometric Distribution Negative Binomial Distribution Poisson Distribution CVE 472 Statistical Techniques in Hydrology. 2/23

3 Hypergeometric Distribution Drawing a random sample of size n (w/o replacement) from a finite population of size N with the elements of the population divided into two groups with k elements belonging to one group. Probability of success (X=x): f x ( x; N, n, k) = k N k x n x N n f x (x;n,n,k): the probability of obtaining X=x success in a sample of size n drawn from a population of size N containing k successes. CVE 472 Statistical Techniques in Hydrology. 3/23

4 Hypergeometric Distribution The mean of the hypergeometric distribution is µ = E(X) = nk N The standard deviation is σ = nk(n-k) 2 N N-n N -1 CVE 472 Statistical Techniques in Hydrology. 4/23

5 Hypergeometric Distribution Examples Solve MS EXCEL Function: =HYPGEOMDIST() Example 4.1 p.69, For a particular watershed, records from 10 rain gages are available. Records from 3 of the gages are known to be bed. If 4 records are selected at random from the 10 records, (a) What is the probability that 1 bad record will be selected? (b) What is the probability that 3 bad records will be selected? (c) What is the probability that at least 1 bad record will be selected? Example 4.2 p.69 CVE 472 Statistical Techniques in Hydrology. 5/23

6 Outline Hypergeometric Distribution Binomial Distribution Geometric Distribution Negative Binomial Distribution Poisson Distribution CVE 472 Statistical Techniques in Hydrology. 6/23

7 Binomial Distribution f x n! x = ( x; n, p) = p (1 p) x!( n x)! n x f x = probability of x successes in n trials, with probability of success p on each trial x = number of successes in sample, (x = 0, 1, 2,..., n) n = sample size (number of trials or observations) p = probability of success CVE 472 Statistical Techniques in Hydrology. 7/23

8 Binomial Distribution The shape of the binomial distribution depends on the values of p and n Mean Here, n = 5 and p = 0.1 Here, n = 5 and p = 0.5 P(X) X P(X) X CVE 472 Statistical Techniques in Hydrology. 8/23

9 Binomial Distribution Mean µ = E(x) = np Variance and Standard Deviation σ 2 = np(1- p) σ = np(1- p) Where n = sample size p = probability of success (1 p) = probability of failure CVE 472 Statistical Techniques in Hydrology. 9/23

10 Binomial Distribution Examples: Solve Example 4.4 (p.71) Example 4.5 (p.72) Example 4.6 (p.72) Example 4.7 (p.72) Example 4.8 (p.73) MS Excel Function =BINOMDIST() CVE 472 Statistical Techniques in Hydrology. 10/23

11 Outline Hypergeometric Distribution Binomial Distribution Geometric Distribution Negative Binomial Distribution Poisson Distribution CVE 472 Statistical Techniques in Hydrology. 11/23

12 Geometric Distribution The probability that the first exceedance (or success) occurs on the x th trial f x = ( x; p) = p (1 p) x 1 x = number of successes in sample, (x = 0, 1, 2,..., n) p = probability of success CVE 472 Statistical Techniques in Hydrology. 12/23

13 Geometric Distribution Mean µ = E(x) = 1/p Variance 2 σ = Var( X ) = (1- p)/p 2 Where p = probability of success (1 p) = probability of failure CVE 472 Statistical Techniques in Hydrology. 13/23

14 Geometric Distribution Examples: Solve Example 4.9 (p.74) Example 4.10 (p.75) CVE 472 Statistical Techniques in Hydrology. 14/23

15 Outline Hypergeometric Distribution Binomial Distribution Geometric Distribution Negative Binomial Distribution Poisson Distribution CVE 472 Statistical Techniques in Hydrology. 15/23

16 Negative Binomial Distribution The probability that the k th exceedance (or success) occurs on the x th trial (x k) f x ( x 1)! k = ( x; k, p) = p (1 p) ( k 1)!( x k)! x k x = number of successes in sample, (x = k, k+1,..., ) p = probability of success Example 4.11: What is the probability that the fourth occurrence of a 10-yr flood will be on the fortieth year? (x=40, k=4, p=1/10) CVE 472 Statistical Techniques in Hydrology. 16/23

17 Negative Binomial Distribution Mean µ = E(x) = k/p Variance σ 2 = Var( X ) = k(1- p)/p 2 Where p = probability of success (1 p) = probability of failure CVE 472 Statistical Techniques in Hydrology. 17/23

18 Outline Hypergeometric Distribution Binomial Distribution Geometric Distribution Negative Binomial Distribution Poisson Distribution CVE 472 Statistical Techniques in Hydrology. 18/23

19 Poisson Distribution Apply the Poisson distribution when you wish to count the number of times an event occurs in a given area of opportunity. f x ( x; λ) = e λ x! λ x where: x = number of events in an area of opportunity (x = 0, 1, 2, ) λ = expected number of events (λ > 0) e = base of the natural logarithm system ( ) CVE 472 Statistical Techniques in Hydrology. 19/23

20 Poisson Distribution Mean µ = λ = np Variance and Standard Deviation σ 2 = λ σ = λ where λ = expected number of events CVE 472 Statistical Techniques in Hydrology. 20/23

21 Poisson Distribution Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameter λ : 0.70 λ = 0.50 λ = P(x) P(x) x x CVE 472 Statistical Techniques in Hydrology. 21/23

22 Poisson Distribution Solve the following examples given in the text book. MS Excel Function Example 4.12 (p.76) =POISSON() What is the probability that a storm with a return period of 20 years will occur in a 10-yr period? Example 4.13 (p.76) What is the probability of 5 occurrences of a 2-yr storm in a 10- year period? Example 4.14 (p.77) What is the probability of fewer than 5 occurrences of a 20-yr storm in a 100-yr period? CVE 472 Statistical Techniques in Hydrology. 22/23

23 Exercises Page 81-84: CVE 472 Statistical Techniques in Hydrology. 23/23