Northeastern University December, 2018

Department of Electrical and Computer Engineering EECE 7204 (Fall 2018)

**Final Exam**

This is a take-home exam that has to be submitted over Blackboard after completion. The due date is December 10* th*, 2018, at 22h.

** Problem 1 (10/100)**: We have two coins:

_{1}and

_{2}. While

_{1}is a fair coin, the second coin has two heads. In an experiment, we select

_{1}with probability 3

*4 and*

*/*_{2}with probability 1

*4. The selected coin is tossed*

*/**times and*

*n**heads are observed.*

*n*(a) Compute the probability of having selected C_{2} given the outcome of the experiment.

(b) What is the limit of that probability when * n *→ ∞ ?

** Problem 2 (10/100)**: Let (

*) be a 2-dimensional random variable with independent mar- ginal components*

*X, Y**and*

*X**which are exponential with expected value*

*Y**=*

*m*_{X}*= 1. Find the probabilities*

*m*_{Y}*(*

*P**≥*

*X**≥ 1) and*

*Y**(*

*P**2*

*X >**|*

*Y**).*

*X > Y*** Problem 3 (10/100)**: Let

*and*

*X**be two uncorrelated random variables and consider their sum*

*Y**=*

*S**+*

*X**and difference*

*Y**=*

*R**. Find a necessary and sufficient condition for*

*X Y**and*

*S**to be also uncorrelated.*

*R*** Problem 4 (10/100)**: Let

*be a continuous random variable uniformly distributed in (0*

*X**1).*

*,*Find the probability density functions of the random variables * Y *=

*(*

*g**) when:*

*X*(a) * g*(

*) = 8*

*x*

*x*^{3}

(b) * g*(

*) = (*

*x**− 1*

*x**2)2*

*/*** Problem 5 (10/100)**: Two friends agree in meeting for a coffee between 9 and 10 in the morning. The times of arrivals at the cafeteria for each one are random and independent, being uniform in that interval. If we denote by

*and*

*X**the time of arrival of each friend, then*

*Y**=*

*T**is the random variable representing the time that the first person has to wait before the other friend arrives. Find E(*

*X Y**), the expected waiting time, using these two methods:*

*T*(a) Compute the cumulative density function of * T *; then obtain the probability density function of

*; and evaluate the expectation.*

*T*(b) Applying the expectation theorem.

** Problem 6 (10/100)**: The random variables

*and*

*X**have a joint probability density function given by*

*Y** fXY *(

*) = . 8*

*x, y**0 ≤*

*xy**≤*

*y**≤ 1*

*x*(a) Obtain the conditional density of * Y *given

*, that is*

*X**(*

*fY |X**).*

*y, x*(b) Compute the conditional expectation E(*Y *^{4} * X *=

*). Discuss why this quantity is a random variable.*

*x*** Problem 7 (10/100)**: Obtain the moment generating function of an exponential random variable

*with parameter*

*X**. Using that result, show that E(*

*µ**) =*

*X*^{n}

*n*__!__

__Hint__: 1 1 * z *= 1 +

*+*

*z*

*z*^{2}+ · · · +

*+ · · · for −1*

*z*^{n}*1*

*< z <*** Problem 8 (10/100)**: Let

*= min(*

*S*_{n}

*X*_{1}

*, X*_{2}

*), where*

*, . . . , X*_{n}

*X*_{k}_{k}_{1}is a sequence of uniform random variables that are independent and identically distributed in the interval [0

*1].*

*,** n−*1

(a) Show that the probability density functions of * S_{n} *is

*(*

*f*_{S}*n**) =*

*s**(1 −*

*n**) in the interval*

*s*0 * < s < *1 and zero otherwise. To do that, first obtain

*(*

*F*_{S}*n**), its probability distribution function.*

*s*(b) Prove that the sequence {* X_{n}*}

_{n≥}_{1}converges to 0 in mean square and in probability.

** Problem 9 (10/100)**:

*and*

*A**are two zero-mean, independent random variables with vari-*

*B*ances *σ*^{2} = *σ*^{2} = 2. Given those random variables, a stochastic process * X*(

*) is defined as*

*t** X*(

*) =*

*t**+*

*At*

*B .*(a) Find the mean and autocorrelation functions of the process * X*(

*). Is it a strict-sense stationary process? Is it a wide-sense stationary process? Justify your answer.*

*t*Assume that * A *and

*are continuous random variables. Show that the first-order probability density function of the process,*

*B**(*

*f*_{X}*;*

*x**), can be expressed in terms of the probability density functions of the random variables as*

*t*where * A *∼

*(*

*f*_{A}*) and*

*a**∼*

*B**(*

*f*_{B}*).*

*b*** Problem 10 (10/100)**: Let

*and*

*A**two uncorrelated, zero-mean random variables with identical variance*

*B*

*σ*^{2}. A stochastic process is constructed such that

*(*

*X**) =*

*t**cos(*

*A*

*ω*_{0}

*) +*

*t**cos(*

*B*

*ω*_{0}

*), where*

*t*

*ω*_{0}is a known constant. Is

*(*

*X**) wide-sense stationary? Is*

*t**(*

*X**) ergodic in the mean?*

*t*