Stat 61 - Homework #5 (10/22/07, replaces draft 10/12/07; due 10/29/07)

 

1.  Let  X  be a random variable with E(X) = 100 and Var(X) = 15.  What are…

            a.  E ( X2 )    (Not 10000)

            b.  E ( 3X + 10 )

            c.  E (-X)

            d.  Standard deviation of  –X ?

 

2.  Let X be the random variable with pdf

 

            fX ( x ) =  2x          if  0 ≤ x ≤ 1

                             0            otherwise.

 

            a.  What is the cdf,  FX(a) ?

            b.  Let  Y  be another random variable defined by  Y = X2.  What is its cdf,

                        FY (b) ?   

( This is possible because if you know P( X ≤ a ) for every a, you can get

P( Y ≤ b ) for any b.)

c.  What is the pdf,  fY ?

            d.  Compute E(Y) directly from FY.  (Use the secret formula.)  (See website if                                necessary)

            e.  Compute E(Y)  from fY.

            f.  Compute E(Y) = E(X2)  directly from fX.  (Part f is the Lazy Statistician’s

                        Rule.  If your answers to d-e-f don’t agree, panic.)

 

3.  (Sum of random variables)  Let  X  and  Y  be independent random variables, with these pmf’s:

 

                        pX(n) = (1/3) for n = 1, 2, 3; else 0

                        pY(n) = (1/2) for n = 1, 2; else 0

 

            Let  Z  =  X + Y.  Using the formula

 

                        ,

 

            construct the pmf for  Z.

 

4.  Now let  X  and  Y  be independent Poisson random variables, with means  mX  and  mY  respectively.  Let  Z = X + Y.  Using the formula from problem 3,  derive the pmf for  Z.

 

            (Hint:  The binomial formula is  . )

 

 

5.  (The mother of all Poisson examples)  Do problem 4.2.10, page 287.  Either actually read the problem and do what it says, or do parts a and b below, which amount to the same thing.  Specifically, given the following distribution…

 

                        k          Observed number of

                                    corps-years in which

                                    k fatalities occurred

                       

                        0          109

                        1          65

                        2          22

                        3          3

                        4          1    

                        ≥5        0    

                                    200     

 

            [This is a frequency table describing 200 annual reports from pre-WWI Prussian cavalry corps, each giving a number of soldiers killed by being kicked by horses.]

 

            It is suggested that these values represent 200 independent draws from a Poisson distribution.

 

            a.  What is a reasonable estimate for the mean of the distribution?

 

            b.  What would the entries in the right-hand column be if the 200 draws were exactly distributed according to this Poisson distribution?  (The entries would not be integers.)

 

            (In November we’ll address this question:  Comparing the original data to the theoretical values in part b, is it plausible that these really are draws from a Poisson distribution?  Or should we discard that theory?)

 

6.  (Editing a Poisson distribution)  At a certain boardwalk attraction, customers appear according to a Poisson process at a rate of  l = 15 customers/hour.  So, if  X  is the number of customers appearing between noon and 1pm,  X has a Poisson distribution with mean 15.

 

            Assume that each customer wins a prize with (independent) probability 1/5. 

 

            Let  Y  be the number of customers winning prizes between noon and 1pm. 

 

            (problem 6 continues on next pageà)

 

 

 

 

            (Problem 6 continued)

 

            One way to understand Y is that if the value of X is given, then Y is binomial, with parameters  n = [value of X]  and  p = 1/5.  This reasoning gives us:

 

                        P ( Y = k ) =

 

            (The second part of the summand is the probability that Y=k given that X=n.)

 

            a.  Simplify this formula, to show that  Y  is itself a Poisson distribution with mean 3. 

 

            (Or, if after getting off to a good start you find this problem too annoying, do

            part b instead.)

 

            b.  Find a much simpler explanation of why Y should be Poisson with mean 3.

 

7.  (Waiting times)  Buses arrive according to a Poisson process with mean (1/10) min-1.  Let  W  be the waiting time from time  t = 0  till the arrival of the second bus. 

 

            a.  What is E(W) ?

 

            b.  Can you construct a  pdf  or  cdf  for W ?  

                        ( Good start:  P(W ≤ t) = 1 – P(exactly 0 buses or exactly 1 bus between times 0 and t).  Another approach:  W is the sum of two independent one-bus waiting times.)

 

8.  (The first boring normal tables problem)  If  Z  is a standard normal variable, what are…

            a.  P ( -1.0 ≤ Z ≤ +1.0 )

            b.  P ( -2.0 ≤ Z ≤ +2.0 )

            c.  P ( -3.0 ≤ Z ≤ +3.0 )

 

9.  (The second boring normal tables problem)  Let  Z  be a standard normal variable.

            a.  If  P ( -a ≤ Z ≤ +a ) = 0.95,  what is  a ?

            b.  If  P ( -b ≤ Z ≤ +b ) = 0.99,  what is  b ?

 

10.  (The third boring normal tables problem)

            a.  If  X  is normal with mean 500 and standard deviation  110, 

                        what is  P ( X ≥ 800 ) ?

            b.  If  X  is normal with mean 500 and standard deviation  120,

                        what is  P ( X ≥ 800 ) ?

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