SampleFinal

SampleFinal - EE 178 Tuesday, March 11, 2008 Probabilistic...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EE 178 Tuesday, March 11, 2008 Probabilistic Systems Analysis Handout #22 Sample Final Problems The following are old exam problems. To prepare for the final you will also need to review the lecture notes, the homework, midterm review, and midterm problems. 1. Inequalities Label each of the following statements with =, , or NONE. Label a statement with = if equality always holds. Label a statement with or if strict inequality is possible. If no such equality or inequality holds in general, label the statement as NONE. Justify your answers. a. P ( n i =1 A i ) vs. 1- 2- 2 n if A 1 ,A 2 ,...,A n are independent and P( A i ) = 3 / 4 for all i . b. E [Var( X + Y | Z )] vs. Var( X ) + Var( Y ) if X and Y are conditionally independent given Z . c. E( X 4 | Y ) vs. (E( X 2 | Y )) 2 . d. E( X 8 ) vs. 1680, if X N (0 , 2). e. M Z ( s ) vs. M X ( s ) M Y ( s ) if Z = X + Y , and X and Y are uncorrelated. f. P { XY < 16 } vs. 3 / 4, if X 0 and Y 0 are independent and E( X ) = E( Y ) = 2. 2. Function of two random variables Let X 1 and X 2 be independent r.v.s, where X 1 is distributed exponentially with param- eter 1 and X 2 is distributed exponentially with parameter 2 . Let B be a Bernoulli r.v., independent of X 1 and X 2 , with P { B = 0 } = P { B = 1 } = 1 2 . Define Z = braceleftbigg min { X 1 ,X 2 } if B = 0 , max { X 1 ,X 2 } if B = 1 . Find the pdf of Z . 3. Function of two random variables Let U and V be two random variables uniformly distributed over the set ( u,v ) such that u 1 and 0 v 1 and let X = UV . a. Find the joint cdf of X and U , F X,U ( x,u ). b. Find and sketch the pdf of X , f X ( x ). c. Find the best MSE estimate of U given X . 4. Additive uniform noise channel Let the random variable = 1 with probabaility 1 / 2 and =- 1 with probability 1 / 2 be the signal transmitted over an additive uniform noise channel with output Y = + Z , where the noise Z U[- 3 / 2 , 3 / 2] is independent of ....
View Full Document

This note was uploaded on 09/29/2008 for the course EE 178 taught by Professor Hewlett during the Spring '08 term at Stanford.

Page1 / 4

SampleFinal - EE 178 Tuesday, March 11, 2008 Probabilistic...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online