Stat 61 - Homework #4 (10/1/07; due 10/8/07)

1.  Mathematics applied to real life.  In roulette, if you bet \$1 on a single number, your probability of winning is 1/38.  You are offered 35-1 odds --- which means that you stand to win \$35 or lose \$1.  Let the random variable X be your profit in dollars from such a bet (so that X can be either +35 or –1).

a.  What is E(X) ?

b.  What is Var(X) ?

c.  What is sX ?

2.  An opportunity to manage risk.  You can also bet on a combination of  L  numbers, where L can be  1, 2, 3, 4, 6, 12, or 18.     Your probability of winning is now  L/38.  If you bet \$1 on this combination, then according to the rules, you stand to win (36/L)-1 dollars or lose one dollar.  (That is, they keep your dollar in any event, but if you win they also give you the payoff of 36/L dollars.)  Let X be your profit from such a bet (so that X can be either  +(36/L)–1   or   –1).

a.  What is E(X) ?

b.  What is Var(X) ?

c.  What is sX ?

3.  Mean and variance for geometric distributions (=Discrete “power laws”).

a.  Let  a  be a constant satisfying  0 < a < 1.  Evaluate:

A = a + 2a2 + 3a3 + 4a4 + …

b.  With  a  as before, evaluate:

B = a + 4a2 + 9a3 + 16a4 + …            (the coefficients are k2 for k = 1, 2, 3, 4…)

c.  Let  µ  be a constant satisfying  µ > 0.  Evaluate:

C = .

d.  Let  µ  be as before.  A random variable  X  has a “geometric distribution with mean µ” if it has the pmf

p(k) = for k = 0, 1, 2, …

(As usual, when p(k) is given only for some values of k, we mean to imply that p(k) = 0 for all other values.  The same goes for densities f(y).)

Find  E(X).

e.  Also find  E(X2).

f.  Also find  Var(X).

g.  Also find   sX,  the standard deviation of  X.

4.  Poisson distributions.  A discrete random variable  X  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, ….

a.  What is E(X) ?

b.  What is Var(X) ?

5.  Mean and Variance for a Standard Normal Distribution.

A random variable  Z  is called “standard normal” and has the “standard normal distribution” if it has the density function

f(z) = for all real z.

a.  First we’ll try to verify that = 1.  Try integration by parts on the integral … write the integrand as and integrate the “(1)” while you differentiate the second factor.  In that way, show that .

Well, that didn’t get us very far, did it?  So let’s save that formula for future reference, and accept on faith (for now) that so that = 1 as required.

b.   Find E(Z).  (Hint:  In the integral, make the substitution u = ½ z2.   (Now you know why we like that “1/2” in the exponent.)

c.   Find E(Z2),  Var(Z),  and   sZ.

6.  The Gamma Function.  The “gamma function” is defined by for every x > 0.

(The definition works for most negative values of x and even for complex values of x, but we only care about positive real values.)  (Integration by parts was good for problem 5, but it’s even better for anything involving gamma functions.)

a.  Show that G(1) = 1.

b.  Show that G(2) = 1.

c.  Show that G(3) = 2.

d.  Graph G(x)  (roughly)  for 0 < x ≤ 3.  (Hint:  G(x) has a unique minimum for x > 0.)

e.  Show that G(x+1) = x G(x)  for all x > 0.   (This is consistent with parts b and c, right?)

f.  Show that G(x) = (x-1)!  if x is a positive integer.

7.  The Gamma Distribution.  Let  a > 0  and let  b  be any real number.  A random variable  Y  has a gamma distribution with parameters  a  and  b  if it has the density function .

(Note that if a = 1,  this is just an exponential distribution with b = 1/m.)

a.  Show that the constant has to be    ba / G(a).

b.  What is E(Y) ?

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