Problem Set 1 with 10 Problems on Random Processes | ECE 534 | Assignments Electrical and Electronics Engineering

ECE 534 RANDOM PROCESSES FALL 2004

PROBLEM SET 1 Due September 8

Please visit the course website soon: courses.ece.uiuc.edu/ece534/

1. Review of Basic Probability

Assigned reading: Chapter one, “Getting started,” of the course notes, including the problems

at the end of the chapter. It would also be good to review some of the final exams from the last

six semesters of ECE413. The past ECE413 homeworks, exams, and their solutions are available

on the ECE 413 website.

Problems to be handed in:

1. Independent vs. mutually exclusive

(a) Suppose that an event Eis independent of itself. Show that either P[E] = 0 or P[E] = 1.

(b) Events Aand Bhave probabilities P[A]=0.3 and P[B]=0.4. What is P[A∪B] if Aand B

are independent? What is P[A∪B] if Aand Bare mutually exclusive?

independent? Could they be mutually exclusive?

2. Conditional probability of failed device given failed attempts

A particular webserver may be working or not working. If the webserver is not working, any attempt

to access it fails. Even if the webserver is working, an attempt to access it can fail due to network

congestion beyond the control of the webserver. Suppose that the a priori probability that the server

is working is 0.8. Suppose that if the server is working, then each access attempt is successful with

probability 0.9, independently of other access attempts. Find the following quantities.

(a) P[ first access attempt fails]

(b) P[server is working |first access attempt fails ]

(d) P[server is working |first and second access attempts fail ].

3. Conditional lifetimes and the memoryless property of the geometric distribution

(a) Let Xrepresent the lifetime, rounded up to an integer number of years, of a certain car battery.

Suppose that the pmf of Xis given by pX(k) = 0.2 if 3 ≤k≤7 and pX(k) = 0 otherwise. (i)

Find the probability, P[X > 3], that a three year old battery is still working. (ii) Given that the

battery is still working after five years, what is the conditional probability that the battery will

still be working three years later? (i.e. what is P[X > 8|X > 5]?)

(b) A certain Illini basketball player shoots the ball repeatedly from half court during practice.

Each shot is a success with probability pand a miss with probability 1 −p, independently of the

outcomes of previous shots. Let Ydenote the number of shots required for the first success. (i)

Express the probability that she needs more than three shots for a success, P[Y > 3], in terms of

p. (ii) Given that she already missed the first five shots, what is the conditional probability that

she will need more than three additional shots for a success? (i.e. what is P[Y > 8|Y > 5])?

(iii) What type of probability distribution does Yhave?

4. CDFs and characteristic function of a mixed type random variable

Let X= (U−0.5)+, where Uis uniformly distributed over the interval [0,1]. That is, X=U−0.5

if U−0.5≥0, and X= 0 if U−0.5<0.

(a) Find and carefully sketch the CDF FX. In particular, what is FX(0)?

(b) Find the characteristic function ΦX(u) for real values of u.

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Download Problem Set 1 with 10 Problems on Random Processes | ECE 534 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

ECE 534 RANDOM PROCESSES FALL 2004

PROBLEM SET 1 Due September 8

Please visit the course website soon: courses.ece.uiuc.edu/ece534/

Review of Basic Probability

Assigned reading: Chapter one, “Getting started,” of the course notes, including the problems at the end of the chapter. It would also be good to review some of the final exams from the last six semesters of ECE413. The past ECE413 homeworks, exams, and their solutions are available on the ECE 413 website.

Problems to be handed in:

Independent vs. mutually exclusive (a) Suppose that an event E is independent of itself. Show that either P [E] = 0 or P [E] = 1. (b) Events A and B have probabilities P [A] = 0.3 and P [B] = 0.4. What is P [A ∪ B] if A and B are independent? What is P [A ∪ B] if A and B are mutually exclusive? (c) Now suppose that P [A] = 0.6 and P [B] = 0.8. In this case, could the events A and B be independent? Could they be mutually exclusive?
1. Conditional probability of failed device given failed attempts A particular webserver may be working or not working. If the webserver is not working, any attempt to access it fails. Even if the webserver is working, an attempt to access it can fail due to network congestion beyond the control of the webserver. Suppose that the a priori probability that the server is working is 0.8. Suppose that if the server is working, then each access attempt is successful with probability 0.9, independently of other access attempts. Find the following quantities. (a) P [ first access attempt fails] (b) P [server is working | first access attempt fails ] (c) P [second access attempt fails | first access attempt fails ] (d) P [server is working | first and second access attempts fail ].
2. Conditional lifetimes and the memoryless property of the geometric distribution (a) Let X represent the lifetime, rounded up to an integer number of years, of a certain car battery. Suppose that the pmf of X is given by pX (k) = 0.2 if 3 ≤ k ≤ 7 and pX (k) = 0 otherwise. (i) Find the probability, P [X > 3], that a three year old battery is still working. (ii) Given that the battery is still working after five years, what is the conditional probability that the battery will still be working three years later? (i.e. what is P [X > 8 |X > 5]?) (b) A certain Illini basketball player shoots the ball repeatedly from half court during practice. Each shot is a success with probability p and a miss with probability 1 − p, independently of the outcomes of previous shots. Let Y denote the number of shots required for the first success. (i) Express the probability that she needs more than three shots for a success, P [Y > 3], in terms of p. (ii) Given that she already missed the first five shots, what is the conditional probability that she will need more than three additional shots for a success? (i.e. what is P [Y > 8 |Y > 5])? (iii) What type of probability distribution does Y have?
CDFs and characteristic function of a mixed type random variable Let X = (U − 0 .5)+, where U is uniformly distributed over the interval [0, 1]. That is, X = U − 0. 5 if U − 0. 5 ≥ 0, and X = 0 if U − 0. 5 < 0. (a) Find and carefully sketch the CDF FX. In particular, what is FX (0)? (b) Find the characteristic function ΦX (u) for real values of u.

Poisson and geometric random variables with conditioning Let Y be a Poisson random variable with mean μ > 0 and let Z be a geometrically distributed random variable with parameter p with 0 < p < 1. Assume Y and Z are independent. (a) Find P [Y < Z]. Express your answer as a simple function of μ and p. (b) Find P [Y < Z|Z = i] for i ≥ 1. (Hint: This is a conditional probability for events.) (c) Find P [Y = i|Y < Z] for i ≥ 0. Express your answer as a simple function of p, μ and i. (Hint: This is a conditional probability for events.) (d) Find E[Y |Y < Z], which is the expected value computed according to the conditional distribu- tion found in part (c). Express your answer as a simple function of μ and p.
Conditional expectation for uniform density over a triangular region Let (X, Y ) be uniformly distributed over the triangle with coordinates (0, 0), (1, 0), and (2, 1). (a) What is the value of the joint pdf inside the triangle? (b) Find the marginal density of X, fX (x). Be sure to specify your answer for all real values of x. (c) Find the conditional density function fY |X (y|x). Be sure to specify which values of x the conditional density is well defined for, and for such x specify the conditional density for all y. Also, for such x briefly describe the conditional density of y in words. (d) Find the conditional expectation E[Y |X = x]. Be sure to specify which values of x this conditional expectation is well defined for.
Functions of independent exponential random variables Let X 1 and X 2 be independent random varibles, with Xi being exponentially distributed with parameter λi. (a) Find the pdf of Z = min{X 1 , X 2 }. (b) Find the pdf of R = X X^12.
Using the Gaussian Q function Express each of the given probabilities in terms of the standard Gaussian complementary CDF Q. (a) P [X ≥ 16], where X has the N (10, 9) distribution. (b) P [X^2 ≥ 16], where X has the N (10, 9) distribution. (c) P [|X − 2 Y | > 1], where X and Y are independent, N (0, 1) random variables. (Hint: Linear combinations of independent Gaussian random variables are Gaussian.)
Uniform density over a union of two square regions Let the random variables X and Y be jointly uniformly distributed on the region { 0 ≤ u ≤ 1 , 0 ≤ v ≤ 1 } ∪ {− 1 ≤ u < 0 , − 1 ≤ v < 0 }. (a) Determine the value of fXY on the region shown. (b) Find fX , the marginal pdf of X. (c) Find the conditional pdf of Y given that X = a, for 0 < a ≤ 1. (d) Find the conditional pdf of Y given that X = a, for − 1 ≤ a < 0. (e) Find E[Y |X = a] for |a| ≤ 1. (f) What is the correlation coefficient of X and Y? (g) Are X and Y independent? (h) What is the pdf of Z = X + Y?
Transformation of densities Let U and V have the joint pdf:

fU V (u, v) =

{ c(u − v)^2 0 ≤ u, v ≤ 1 0 else

for some constant c. (a) Find the constant c. (b) Suppose X = U 2 and Y = U 2 V 2. Describe the joint pdf fX,Y (x, y) of X and Y. Be sure to indicate where the joint pdf is zero.

Problem Set 1 with 10 Problems on Random Processes | ECE 534, Assignments of Electrical and Electronics Engineering

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Download Problem Set 1 with 10 Problems on Random Processes | ECE 534 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

ECE 534 RANDOM PROCESSES FALL 2004