Various Practice Problems
1. A student and a professor agree to meet in the MSEE atrium at 2pm to go over the homework. Let X be the arrival time of the student, and let Y be the arrival time of the professor. Assume that the 2D random variable (X,Y) is uniformly distributed in the square [2 , 3]x[2,3].
If the student arrives first, then he will wait up to 20 minutes for the professor to arrive: if the professors does not show up within that time frame, then the student will leave.
The Professor is more impatient: she will leave if she has to wait for more than 10 minutes for the student to arrive.
What is the probability that the meeting will occur?
2. Let (X,Y) be a 2D random variable that is uniformly distributed in the rectangle [1,3]x[5,10].
Find the conditional probability density function
$ f_{X|Y}(x|7). $
3. Let (X,Y) be a 2D random variable that is uniformly distributed inside the ellipse defined by the equation
$ (\frac{x}{a})^{2}+(\frac{y}{b})^{2}=1, $
for some constants a,b>0. Find the conditional probability density function $ f_{X|Y}(x|y). $
4. Let X be a continuous random variable with probability density function
$ f_X(x)=\left\{ \begin{array}{ll} c x^2, & 1<x<5,\\ 0, & \text{ else}. \end{array} \right. $
Let A be the event $ \{ X>3 \} $. Find the conditional probability density function $ f_{X|A}(x|A). $
5. Two continuous random variables X and Y have the following joint probability density function:
$ f_{XY} (x,y) = C e^{\frac{-(4 x^2+ 9 y^2)}{2}}, $
where C is an appropriately chosen constant. Are X and Y independent? Answer yes/no and give a mathematical proof of your answer.
6. A discrete random variables X has a moment generating (characteristic) function $ M_X(s) $ such that
$ \ M_X(j\omega)= 3+\cos(3\omega)+ 5\sin(2\omega). $
Find the probability mass function (PMF) of X.
7. Let X be an exponential random variable. Recall that the pdf of an exponential random variable is given by
$ \ f_X(x)= \lambda e^{-\lambda x}, \text{ for }x\geq 0 . $
Obtain the moment generating function $ M_X(s) $ of X.
8. Let X be a continuous random variable with pdf $ f_X(x) $. Let $ Y=aX+b $ for some real valued constants a,b, with $ a\neq 0 $. What is the pdf of the random variable Y?
9. Let
$ X=\left( \begin{array}{l} X_1\\ X_2 \end{array} \right) $
be a two-dimensional Gaussian random variable with mean $ \mu $ and standard deviation matrix $ \Sigma $ given by
$ \mu=\left( \begin{array}{c} -1\\ 2 \end{array} \right) , \Sigma=\left( \begin{array}{cc} 3 & 1 \\ 1 & 3 \end{array} \right) $
a) Write the pdf of X using matrix notation.
b) Write the pdf of X without matrix or vector.
c) Find the marginal pdf for $ X_1 $.
d) Find a matrix M such that the vector $ Y=M(X-\mu) $ consists of independent random variables.
e) Find the joint pdf of Y.
