Geometric Distribution

It's about geometric distribution which is a discrete probability distribution.

1. Discrete Probability Distribution Geometric Distribution 1 Geometric Distribution

2. Topics of Geometric Distribution Definition and PDF. Examples. Uses. Properties Necessary . Applications. Mean and Variance. Moment Generating Function. Skewness And Kurtosis. Geometric Distribution 2

3. Definition of Geometric Distribution The Geometric Distribution a discrete probability distribution. It is the distribution of the number of trials needed to get the first success in repeated Bernoulli trials. Satisfies the following conditions: A trial is repeated until a success occurs. The repeated trials are independent of each other. The probability of success p is constant for each trial. x represents the number of the trial in which the first success occurs. Geometric Distribution 3

4. PDF of Geometric Distribution Probability of the 1 st success on the N th trial, given a probability, p, of success Geometric Distribution 4 To show P(N=x) is a proper pdf :

5. Example of Geometric Distribution You play a game of chance that you can either win or lose  until you lose. Your probability of losing is p=0.57. What is the probability that it takes five games until you lose? Geometric Distribution 5 Solution: • Geometric with p = 0.57 , q = (1-p) = 0.43 , x = 5 P(x=5) = = 0.57(0.43) 5-1 = 0.0194871657

6. Uses of Geometric Distribution In sports, particularly in baseball, a geometric distribution is useful in analyzing the probability. In cost-benefit analyses, such as a company deciding whether to fund research trials. In time management, the goal is to complete a task before some set amount of time . Geometric Distribution 6

7. Properties of Geometric Distribution The mean or expected value of a distribution gives useful information about what average one would expect from a large number of repeated trials. The median of a distribution is another measure of central tendency, useful when the distribution contains outliers (i.e. particularly large/small values) that make the mean misleading. The mode of a distribution is the value that has the highest probability of occurring. The variance of a distribution measures how "spread out" the data is. Related is the standard deviation--the square root of the variance--useful due to being in the same units as the data . Geometric Distribution 7

8. Necessary of Geometric Distribution Geometric distribution used to model probability. Which is very important in statistics and in everyday life. Businesses , governments, and families use to make important decisions. Geometric distributions provide statistical models that show the possible outcomes of a particular event. These models give people the ability to make decisions. Geometric Distribution 8

9. Applications of Geometric Distribution used in Morkov chain models, particularly meteorological mode of weather cycles and precipitation amounts. referred to as the failure time distribution. used to describe the number of interviews that have to be conducted by a selection board to appoint the first acceptable candidate. Geometric Distribution 9

10. Mean And Variance of Geometric Distribution The Mean of the Geometric distribution is: E(x) = pq x-1 = p + 2pq + 3pq 2 + 4pq 3 + ………….. = p(1 + 2q + 3q 2 + 4q 3 + ………) = p(1-q) -2 =   Geometric Distribution 10

11. Mean And Variance of Geometric Distribution The Variance of the Geometric distribution is : E(X 2 )= pq x-1 = p + 4pq + 9pq 2 + 16pq 3 +……………. = p(1 + 4q + 9q 2 + 16q 3 +…………….) = p(1 + 3q + 6q 2 + 10q 3 +…..) + pq (1 + 3q + 6q 2 +……..) = p (1-q) -3 + pq (1-q ) -3 = + Hence, Variance= µ 2 = E(x 2 ) - [E(x)] 2 = + - =   Geometric Distribution 11

12. Moment Generating Function of Geometric Distribution Moment Generating Function of Geometric Distribution is given bellow: M X (t) = E( e tx ) = = pq x = p = p + pqe t + pq 2 e t2 + pq 3 e t3 +………… = p[1 + qe t + ( qe t ) 2 + ( qe t ) 3 + ………..] = p(1-qe t ) -1 =   Geometric Distribution 12

13. Moments of Geometric Distribution The moments of Geometric Distribution are given bellow: µ’ 1 = µ’ 2­ = µ’ 3 = µ’ 4 =   Geometric Distribution 13

14. Central Moments of Geometric Distribution The 2 nd , 3 rd and 4 th central moments of Geometric Distribution are given bellow: µ 2 = µ’ 2 – (µ’ 1 ) 2 = µ 3 = µ’ 3 - 3µ’ 2 µ’ 1 + 2(µ’ 1 ) 3 = µ 4 = µ’ 4 - 4µ’ 3 µ’ 1 + 6µ’ 2 (µ’ 1 ) 2 - 3(µ’ 1 ) 4 =   Geometric Distribution 14

15. Shape Characteristic of Geometric Distribution The S kewness and Kurtosis of Geometric Distribution are given bellow: 1 = = Kurtosis 2 = =   Geometric Distribution 15

16. Thank You Have a good day Geometric Distribution 16