Important Probability Distributions Discrete Probability Distributions Geometric Distribution Remarks: The geometric distribution is the simplest of the waiting time distributions.The random variable x is the num- ber of trials required to obtain the first success,so we are“waiting for a success'”. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 26/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 26/135 Geometric Distribution Discrete Probability Distributions Remarks:
Important Probability Distributions Discrete Probability Distributions Geometric Distribution Remarks: The geometric distribution has the so-called "memory- less”or“nonaging'”property:for integers s>t,we have P(X>s X>t=P(X>s-t). That is,the probability of getting additional s-t failures, having already observed t failures,is the same as the probability of observing s-t failures at the start of the sequence.In other words.the probability of getting a run of failures depends only on the length of the run,not on its position. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 27/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 27/135 Geometric Distribution Discrete Probability Distributions Remarks:
Important Probability Distributions Probability mass function 1.0 1.0 0p=0.2 p=0.2 0.8 。p=0.5 0.8p p=0.5 0 p=0.8 p=0.8 ×0.6 ×0.6 a0.4 0.4 0.2 0.2 0.0 0.0 10 0 8 10 Cumulative distribution function 1.0 1.0 0.8 0 0.8 ×0.6 00 ×0.6 o ●0 。p=0.2 o p=0.2 0.2 ·p=0.5 0.2b-o p=0.5 °p=0.8 0 p=0.8 0.0 0.0 0 6 8 10 0 6 8 10 Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 28/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 28/135 Probability mass function Cumulative distribution function
Important Probability Distributions Discrete Probability Distributions Geometric Distribution Question:How to show the "memoryless"property? o I Using the formula P(A B)=P(AB)/P(B)and the fact that when s >t,the event >t contains the event>s as a subset,we have To be Continued Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 29/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 29/135 Geometric Distribution Discrete Probability Distributions To be Continued
Important Probability Distributions Discrete Probability Distributions Geometric Distribution P(X>s,X>t) P(X>s X>t= PX>t) P(X>s) 三 P(X>t) 1-P(X≤S) 三 1-P(X≤t) 1-∑=1p(1-p)z- 三 1-∑=1p1-p)x-1 (1-p) (1-p) (1-p)8-t 三 P(X>s-t). To be Continued Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 30/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 30/135 Geometric Distribution Discrete Probability Distributions To be Continued