Advanced Econometrics (Econ 211a)
Problem Set 1
(Due Thursday, 10/1)
Theoretical Problems
1. Let X be a continuous random variable with the following probability density function (p.d.f.):
f(x)= 1/2
0
for 1  x  3
otherwise
(a) Calculate E[X].
(b) Compute Var[X].
(c) Find Prob(1  X  3/2).
2. Let X represent the number of times tails comes up in 3 tosses of a coin.
(a) Write out the form of the pdf and cdf for this random variable. Is it a discrete or continuous
random variable?
(b) Graph the pdf and cdf for this random variable.
(c) Show that these represent a valid pdf and cdf by examining the axioms of probability.
(d) Calculate Prob(X > 1), Prob(X >= 1), and (Prob 1 < X < 3).
(e) Calculate the mean and variance of this random variable.
3. The cumulative distribution function (cdf) for a random variable, X, is:
F(x) =
1
1 + e- x
(a) Find the pdf for this random variable. Is it a discrete or continuous random variable?
(b) Graph the pdf and cdf for this random variable. You might want to use a spreadsheet program to
graph it.
(c) Show that these represent a valid pdf and cdf.
(d) Calculate Prob(-1 < X < 2) and Prob (X > 1).
4. Suppose that the p.d.f. of a certain random variable X has the following form:
f(x)
= cx
=0
for 0 < x < 4,
otherwise,
where c is a given constant. What are the values of c, Prob(1 <= X <= 2), and
Prob(X > 2)?