Stat 61 - Homework #3 (9/26/07; due 10/1/07)

 

1.  Roll two standard dice.  Let X = larger of the two numbers showing. 

 

                a.  Construct a table showing both the probability mass function (pmf)  pX  and the  cumulative density function (cdf)  FX.

 

                b.  What is FX(7) ?                (Note:  “pmf” = what the text calls a “pdf” in the case of a discrete r.v.)

 

2.  Let  Y  be a binomial random variable with parameters  n = 6  and  p = 1/2.

 

                a.  What is  P ( Y = 3 ) ?

 

                b.  What is  P ( Y > 3 ) ?  [Can you do part b with zero extra work?]

 

3.  Z is a random variable with the pmf

 

                                pZ (k)  =  C (1/10)k     if  k = 1, 2, 3, 4, …

                                                (or zero if n has any other value)

                                                (in particular, pZ(0) = 0)

 

                …where C is a particular real number.  What is  C ?

 

4.  A bar of Uranium contains  n  atoms.  Each of them has (independent) probability  p  of decaying during a one-minute test.  Let  W  be the random variable whose value is the number of atoms that decay during the one-minute test.

 

                What is  P ( W = 2 )  if…

 

                a.  n = 2  and  p = 0.5

                b.  n = 10  and  p = 0.1

                c.  n = 100  and  p = 0.01

                d.  n = 1023  and  p = 10-23  (approximate if you must)

 

5.  A discrete random variable has a Poisson distribution with parameter m  and is called a Poisson random variable with parameter  m  if its pmf is given by

 

                                p(k) = e-m ( mk / k! )     if  k = 0, 1, 2, 3, …

                                                or 0 for any other value of k.

 

                Show that (as we would hope) 

 

                                p(0) + p(1) + p(2) + …  = 1.

 

                (This is really a Math 25 problem.  Think of a Taylor series.)

 

6.  (from text)  A continuous random variable  X  has the following  cdf:

 

                                FX (a)  =  0    if  a ≤ 0

                                                 a2  if  0 ≤ a ≤ 1

                                                 1   if   a ≥ 1.

 

                a.  What is  P ( 1/2 ≤ X ≤ 3/4 ) ?

 

                b.  What is the probability density function (pdf) for  X ?

 

                c.  Express the answer to (a) as an integral involving the pdf.

 

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