Download ***Rice, problem 4, page 154

C22.0015 / B90.3302
Homework Set 1
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1. Rice, problem 4, page 166.
You should distinguish the cases “E X does not exist” from “E X = ” as these
have slightly different meanings. In this problem X is positive, so that the “E X
does not exist” case cannot happen. For some values of , we get finite E X,
while for other values of  we have E X = . Be sure to identify the appropriate
ranges for the  values.
2. Rice, problem 6, page 167.
3. Rice, problem 15, page 168. Assume for simplicity that each lottery has one winning
ticket with prize M and n – 1 losing tickets. You can set aside c, the cost of a ticket, as
a sunk cost.
4. This is an extension of Rice’s problem 15, page 168. In that problem you showed (we
hope!) that the total expectation for two tickets is the same, regardless of whether the two
tickets are from the same lottery of from different lotteries. In terms of M and n, find
Var(X + Y), where X and Y are the payoffs from purchasing one ticket in each of two
different lotteries. Then find Var(Z1 + Z2), where Z1 and Z2 are the payoffs from
purchasing two tickets in the same lottery. As before, we can set aside the value c as a
“sunk cost” so that it will be convenient to have random variables whose only values
are 0 and M.
5. Suppose that random variables X and Y are uniformly distributed over the triangle
with corners at (0, 0), (0, 10), and (10, 0).
(a)
(b)
(c)
(d)
(e)
(f)
Give the joint density function fX, Y (x, y).
Give the marginal density of X. Call it fX (x). Along with this, give E(X)
and Var(X).
Give the marginal density of Y. Call it fY (y). Why should this be an
easy problem?
Give the conditional density of Y, given X. Call this fY | X (y | x).
Find the conditional expectation E(Y | X) and the conditional variance
Var(Y | X).
Verify that Var(Y) = E{ Var (Y | X ) } + Var{ E( Y | X ) }.
6. The random variable X is Poisson with mean . Conditional on X = x, the random
variable Y is binomial (x, p).
Just to keep things very clean, we’ll make the side note that P[ Y = 0 | X = 0 ] = 1.
(a)
Find the marginal distribution of Y.
(b)
Find the conditional distribution of X | Y = y.
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C22.0015 / B90.3302
Homework Set 1
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7. The random variable Q is normally distributed with mean  and with standard
deviation .
(a)
If  = 40 and  = 8, find P[ Q > 50 ].
(b)
The conditional distribution of X, given Q = q, is normal with mean 0.1 q and
standard deviation 10. Based on a sample of n values, the conditional
distribution of X , given Q = q, is normal with mean 0.1 q and standard deviation
10
. Find the marginal distribution of X . Do this first in terms of general
n
symbols , , and n. Then give this for the case  = 40,  = 8, and n = 25.
HINT: The joint distribution of (Q, X ) is bivariate normal.
(c)
Suppose that you have observed X = x . (This means that the random variable
X has been observed, and its value is x .) Find the conditional distribution of Q,
given X . Do this first in terms of general symbols , , and n. Then give this
for the case  = 40,  = 8, and n = 25.
X
HINT: It helps to use a fact from bivariate distributions. If   has a
Y 
 
bivariate normal distribution with mean  X  and variance matrix
 Y 
 2X
 X Y 

 , then E(Y | X = x) = 0 + 1 x, where
2



X
Y
Y


Y
1 = 
and 0 = Y  1  X . With substitutions, this is
X

E(Y | X = x) = Y   Y  x   X  .
X
In addition, Var(Y | X = x) = Y2 1  2  .
(d)
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Calculate P[ Q > 50 | X = 5.2 ]. Do this for the case  = 40,  = 8, and
n = 25.
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