Exam P

Exam P  –  Distributions  Distributions and Other Useful Formulas: Page 1 Distribution Name Binomial (Bernoulli if )



Hypergeometric

   

                                                                                                              Parameters and Domains

 sample size  population size  subgroup size

Poisson

Geometric (number of failures before first success) Geometric (number of trials including first success)  Negative Binomial (failures before th success)





 and an other important facts Approximation to  hypergeometric if 





  

Items are chosen with replacement

      

Very similar to  binomial:  and

 is the finite

 population correction factor Items are chosen without replacement Approximation to  binomial for large large  and 

small 



  

         



 Negative binomial binomial with Memoryless

 Negative binomial binomial with  (for alternate form) Memoryless For number of trials including the th success, define . 



Exam P  –  Distributions and Other Useful Formulas: Page 2

   

Distribution Name

Uniform (Discrete)

Uniform (Continuous)

Exponential (Gamma with )



Gamma



 form:

                         form:

 form:

 

 Normal (

  

)

          

Parameters and Domains





 

 form:



 

 form:

                                    and an other important facts

 

 form:

Memoryless

  

Median = Waiting time for Poisson events

                                form:

 form:

 form:

 form:

 form:

 form:

 and  is the standard normal

Exam P  –  Distributions and Other Useful Formulas: Page 3 Distribution Name

Lognormal

Pareto Weibull

Beta Chi-Squared

Bivariate  Normal

                                    Parameters and Domains

Let

,

A complete mess. See  parameters and domains column.

Gamma with  and ; has

           Let

,

. Then

                                                           form:

 form:

 form:

 form:

 form:

 form:

                  and an other important facts

Let Then if

,

.





        , df = 1; for

If

,

Switch the variables around to get the conditional expected value and variance for  and .

Exam P  –  Distributions and Other Useful Formulas: Page 4 Multinomial ( variables) Bivariate Uniform

                                Let  be the region in the

-plane. Then

. If  is a rectangle with sides parallel to the axes in

, then  and  are independent and uniformly distributed in

respectively.

    and

,