This exam covers pretty much the same material as our upcoming exam, with the one difference that variance and covariance will NOT be on our exam. Solutions to the practice exam are posted below the exam. Please try to first do the exam WITHOUT looking at the solutions, and only then check your answers.

**Problem 1. Pdf's vs. cdf's**

A random variable *X* has the *cumulative* distribution function

a) Is *X* a discrete random variable, a continuous random variable,
or some sort of mixture?

b) Find the probability density function .

c) What is the probability that ? Simplify your answer as much as possible. (Do NOT leave it as some sort of sum or integral).

**Problem 2. Expectations and Variances**

Let *X* be a continuous random variable with pdf

a) Compute the mean *E*(*X*) and the variance *Var*(*X*).

b) Let
,
and
be independent random variables, each with the same distribution function
as *X*. Let
. Compute the mean and variance of *Y*. (Note: You get part credit
for expressing *E*(*Y*) and *Var*(*Y*) in terms of
*E*(*X*) and
*Var*(*X*), even if you couldn't do part
(a))

**Problem 3. Some standard distributions**

We flip a fair coin until it comes up heads. Let *Y* be the number
of flips before it first does so (that is, *Y*=1 if the first toss
is heads, *Y*=2 if the first is tails and the second is heads, and
so on) Meanwhile, let *X* be a Poisson random variable with mean 3.

a) Write down the pdf's and .

b) Assuming *X* and *Y* are independent, write down the joint
pdf
.

c) What is the probability that *X*+*Y*=13? You may leave
your answer as a sum - simplify *each term* in the sum as much as
possible, but DO NOT attempt to do the whole sum.

**Problem 4. Joint random variables (2 PAGES!!!!)**

Let *X* and *Y* be discrete random variables with a joint
pdf given by the following table:

(The conversion to HTML left out the key to the table. The numbers
across the top refer to X, while

those along the left refer to Y. So, for example, the probability
that (X=-1 and Y=1) is 7/45.)

[Note: Before starting this problem, you may wish to compute the marginal pdf's and ]

a) Are *X* and *Y* independent random variables? Are the events
*X*=0 and *Y*=1 independent? What about the events *X*=-1
and *Y*=-1?

b) Compute *P*(*X*=1|*Y*=-1), the (conditional) probability
of *X*=1, given that *Y*=-1.

c) Compute *E*(*X*), *E*(*Y*) and *E*(*XY*).
From this, compute the covariance of *X* and *Y*.

*Problem 1. Pdf's vs. cdf's*

*A random variable X has the cumulative distribution function*

*a) Is X a discrete random variable, a continuous random variable,
or some sort of mixture?*

Since is continuous, we are dealing with a CONTINUOUS random variable.

*b) Find the probability density function
.*

.

*c) What is the probability that
? Simplify your answer as much as possible. (Do NOT leave it as some sort
of sum or integral).*

(also equals 1/2 - 1/(1+*e*)). Since X is continuous,
we don't have to worry about the difference between "less than" and "less
than or equal". The probability that X is exactly 0, or exactly 1,
is zero. You could also get the answer by integrating f(t)
from 0 to 1.

**Problem 2. Expectations and Variances**

*Let X be a continuous random variable with pdf*

*a) Compute the mean E(X) and the variance Var(X).*

. . .

*b) Let
,
and
be independent random variables, each with the same distribution function
as X. Let
. Compute the mean and variance of Y. (Note: You get part credit for expressing
E(Y) and Var(Y) in terms of E(X) and
Var(X), even if you couldn't do part
(a))*

. .

*Problem 3. Some standard distributions*

*We flip a fair coin until it comes up heads. Let Y be the number
of flips before it first does so (that is, Y=1 if the first toss is heads,
Y=2 if the first is tails and the second is heads, and so on) Meanwhile,
let X be a Poisson random variable with mean 3.*

*a) Write down the pdf's
and
.*

, ; , .

*b) Assuming X and Y are independent, write down the joint pdf
.*

Since *X* and *Y* are independent,
for
,
.

*c) What is the probability that X+Y=13? You may leave your answer
as a sum - simplify each term in the sum as much as possible, but DO NOT
attempt to do the whole sum.*

. Note that we only get contributions for *n* up through 12, since
*n*=13 implies *m*=0, and *m* is always at least 1.

*Problem 4. Joint random variables (2 PAGES!!!!)*

*Let X and Y be discrete random variables with a joint pdf given by
the following table:*

*a) Are X and Y independent random variables? Are the events X=0 and
Y=1 independent? What about the events X=-1 and Y=-1?*

No, yes, and no. *P*(*X*=0 and *Y*=1)= 8/45 = *P*(*X*=0)
*P*(*Y*=1), but
*P*(*X*=-1 and
. It just takes one counterexample to ruin independence, so *X* and
*Y* are NOT independent.

b) Compute *P*(*X*=1|*Y*=-1), the (conditional) probability
of *X*=1, given that *Y*=-1.

*P*(*X*=1 and *Y*=-1)/*P*(*Y*=-1) = (3/45)/(6/45)
= 1/2

c) Compute *E*(*X*), *E*(*Y*) and *E*(*XY*).
From this, compute the covariance of *X* and *Y*.

*E*(*X*) = -12/45 + 0*(15/45) + 1*(18/45) = 6/45 = 2/15
*E*(*Y*)
= -6/45 + 0*(15/45) + 1*(24/45) = 18/45 = 2/5
*E*(*XY*) = 1/45
- 3/45 - 7/45 + 9/45 = 0, so
*Cov*(*X*,*Y*) = *E*(*XY*)-*E*(*X*)*E*(*Y*)
= -4/75.