Stat 330
Homework 4
Spring 2009
This assignment is for extra credit. Please turn in this assignment by Friday, February 13 for 5 extra credit
points added to your homework score at the end of the semester.
1
Probability Mass Functions
Let X be a random variable with image Im(X) = {0, 1, 2, 3}.
(a) Fill in the blanks in the table below to make it a valid probability mass function:
x
pX (x)
0
0.5
1
2
0.25 0.1
3
(b) Derive the distribution function for X and draw it in a chart.
(c) Determine the probabilities that...
(i) X is at least 2.
(ii) X is neither 0 nor 2.
(iii) X is non-negative.
(d) Find the expected value and variance of X.
(e) Let Y be a random variable with Y = 5 − 2X. Determine the image of Y . Based on the rules for
expected values and variances, find the expected value and variance of Y .
2
Calculus with Variances
Based on the definition of variances and expected values, show that the following two statements are true:
(a) V ar(aX) = a2 V ar(X) for any value a ∈ R
(b) V ar(X − Y ) = V ar(X) + V ar(Y ) for two independent random variables X and Y . You may use the
property that V ar(X + Y ) = V ar(X) + V ar(Y ) for two independent random variables X and Y .
3
Market Shares, Baron p.67
Shares of company A cost $10 per share and give a profit of X%. Independently of A, shares of company B
cost $50 a share and give a profit of Y %. Deciding how to invest $1,000, you decide between three portfolios:
(i) 100 shares of A,
(ii) 50 shares of A and 10 shares of B,
(iii) 20 shares of B.
The probability mass functions of X and Y are given as
x
-3
0
3
pX (x) 0.3 0.2 0.5
y
pY (y)
-3
0.4
0
0
3
0.6
(a) Compute expected values and variances of the total dollar profit generated by each portfolio.
(b) If our goal is to maximize our expected value and minimize our variance, what is the least risky/ most
risky portfolio?