Download Homework #7 - Bryn Mawr College

Math 295
Homework #7 (parts 1 and 2)
(due October 28, 2002)
Problem A. A card is drawn at random from a deck of six cards, numbered 2 through 7.
Let X = the number on the selected card.
a. Construct a probability function for X, pX( ).
b. What is E(X) ?
Problem B. Consider this experiment: Roll two three-sided dice. Use this sample space…
11 12 13
21 22 23
31 32 33 …with uniform probabilities.
a. Let Y = sum of the two dice. What is E(Y)?
b. Let Z = absolute difference of two dice (highest minus lowest, regardless of sign, so
that Z is never negative). What is E(Z)?
c. Let W = Y plus Z. That is, for every s in the sample space, W is defined by
W(s) = Y(s) + Z(s). What is E(W)?
d. Let U = Y times Z. What is E(U)? Is it the same as E(Y)E(Z)?
e. Let V = the square of the number on the first die (regardless of the second die). What
is E(V)?
f. Let T = Y times V. What is E(T)? Is it the same as E(Y)E(V)?
Note: It might help to make one large table showing the values of Y, Z, W, U, V, and T
for each outcome in the sample space. Once you have done this, is it worth the
trouble to construct a pmf for each random variable, or not?
Here are the secrets:
(1) If W = X + Y, then E(W) = E(X) + E(Y) always, provided only that E(X) and E(Y) exist. This is
true regardless of what X and Y are, or how they are related.
(2) If U = YZ, then E(U) may or may not be the same as E(Y)E(Z). We will discover when these numbers
are equal. When they are not equal, we will make a big deal of the difference E(Y)E(Z) – (YZ).
Problem C. Y has density function
2y3
if y  1
f (y)  
otherwise.
 0
2
0
a.
b.
c.
d.
1
Is this really a possible density function? (Verify that its integral is 1)
What is the cdf of Y?
What is P( 1  Y  2 ) ?
What is E(Y) ?
Problem D. If the random variable X has the probability function
0
 k:

1
pX (k) : 3
1
2
1
1
3
6
4
,
1
6
determine the expected value, variance, and standard deviation of X.
Problem E. Let X be a random variable with E(X) = 100 and Var(X) = 15. What are…
a. E(X2)
b. E(3X+10)
c. E( – X)
d. Var ( – X )
e. Standard deviation of –X
Problem D. If Y is a random variable that takes values in [-1,+1] and has density
fY(y) = (3/2)y2
in that range, then what are (X) and 2(X) ?