Download Probability Quiz - Random Processes | ECE 534 and more Quizzes Electrical and Electronics Engineering in PDF only on Docsity!
University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering ECE 434: Random Processes Spring 2004 Probability Quiz
Wednesday, February 11, 5:00–7:00pm, 165 Everitt Laboratory
READ THESE COMMENTS BEFORE STARTING THE EXAM!
- This is a closed-book exam! Calculators should not be necessary, but feel free to use one.
- Write your name on the answer booklet.
- There are five unequally weighted problems for a total of 50 points. A bonus problem worth 5 points is also included. Problems are not necessarily in order of difficulty.
- A correct answer does not guarantee credit; an incorrect answer does not guarantee loss of credit. Provide clear explanations, show all relevant work and justify your answers! If we cannot make sense of your writing or reasoning, you may loose points.
- Read each problem carefully and think before performing detailed calculations.
- Only the supplied answer booklet is to be handed in. No additional pages will be considered in the grading. You may want to work things through in the blank areas of the exam and then neatly transfer to the answer sheet the work you would like us to look at.
Problem 1 (10/50, equally weighted parts)
This problem has five independent true/false questions.
(a) Consider three random variables X, Y and Z with joint pdf
fX,Y,Z (x, y, z) =
{ 4 e−z(x+y), 0 ≤ x, y ≤ +∞, 2 ≤ z ≤ 4 , 0 , else.
Then, X and Y are independent when conditioned on Z.
(b) If two random variables X and Y are uncorrelated (i.e., if E[XY ] = E[X]E[Y ]) then Var(X + Y ) = Var(X) + Var(Y ).
(c) If A and B are independent then P (A ∪ B) = P (A) + P (B).
(d) Let A and B be two independent events on a sample space Ω so that P (A) = P (B) 6 = 0 and P (A ∪ B) = 32 P (A). Then, P (A) = 12.
(e) Let X be a Gaussian random variable with zero mean and variance σ^2 , i.e.,
fX (x) =
2 πσ
e−^
x^2 2 σ^2 ,
and let Y = g(X) be another random variable defined by the continuous function g
g(x) =
{ 0 , x ≤ 0 , x^2 , x > 0.
Then, Y is a continuous random variable.
Problem 3 (10/50, unequally weighted parts)
Let X be a random variable with density
fX (x) =
{ αe−x, x ≥ b, 0 , else.
(a) (3 Points) Find α such that fX (x) is a proper density function.
(b) (3 Points) Find the mean of X.
(c) (4 Points) Let Y be another random variable jointly distributed with X. The joint distribution is unknown but the marginal density of Y is the Gaussian distribution i.e.,
fY (y) =
√^1
2 πσ
e−^
(y−μ)^2 2 σ^2.
With the given information, can the following quantities be calculated: (i) E[X + Y ], (ii) E[XY ], (iii) E[X|Y = 1]? If a quantity is calculable, calculate it; otherwise explain why.
Problem 4 (15/50, equally weighted parts)
Random variables X and Y have joint pdf fX,Y (x, y) that is constant in the shaded region (and zero elsewhere).
y
1
1
x
(a) Determine and sketch the densities fX (x) and fY (y).
(b) Are X and Y statistically independent? Explain.
(c) Determine E[X|Y = y] for 0 ≤ y ≤ 1.
(d) Let Z be another Random variable defined as Z = ey. Determine E[X|Z = z] for 1 ≤ z ≤ e.
(e) Let U be another random variable defined as U = X + Y. Determine the density fU (u).
Bonus Problem (5/50)
Prove or disprove the following statement: Given a probability space (Ω, F, P) with Ω contain- ing N elements, then the σ-field F contains necessarily 2n^ elements for some integer 0 < n ≤ N.
