fall09finalsoln

fall09finalsoln - University of Illinois Fall 2009 ECE 313:...

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: University of Illinois Fall 2009 ECE 313: Final Exam Tuesday, December 15, 2009, 8:00 a.m. 11:00 a.m. 100 Materials Science and Engineering Building 1. (a) A , B , and C are events such that 0 < P ( A ) < 1, 0 < P ( B ) < 1, and 0 < P ( C ) < 1. P ( A B ) = P ( A c B c )- P ( A c )- P ( B c ) + 1 is a TRUE statement. Note that the right side is- P ( A c B c )+1 and the result follows from DeMorgans theorem. P ( AB ) P ( A ) + P ( B )- 1 is a TRUE statement. Transposition gives P ( A ) + P ( B )- P ( AB ) = P ( A B ) 1 which obviously is true. P ( A c | B ) P ( B ) + P ( A c | B c ) P ( B c ) = P ( A c ) is a TRUE statement. It is just the law of total probability applied to P ( A c ). P ( A c | B ) P ( B ) + P ( A | B ) P ( B ) = P ( B ) is a TRUE statement. Note that P ( B ) is a common factor on the left side and P ( A c | B ) + P ( A | B ) = 1, P ( A c | B ) P ( B ) + P ( A | B c ) P ( B c ) = P ( A B )- P ( AB ) is a TRUE statement. Both sides equal P ( A B ). P ( B | A ) = P ( A | B ) P ( A ) /P ( B ) is a FALSE statement. The similar-looking P ( B | A ) = P ( A | B ) P ( B ) /P ( A ) is, of course, just Bayes formula. If A and B are mutually exclusive events, then they are independent events is a FALSE statement. Mutually exclusive events are independent only in the trivial case when at least one of the events has zero probability. If A , B , and C are independent events, then P ( ABC ) = P ( A ) P ( B ) P ( C ) is a TRUE statement. The condition P ( ABC ) = P ( A ) P ( B ) P ( C ) is one of four conditions that must hold for A , B , and C to be called independent events. If P ( ABC ) = P ( A ) P ( B ) P ( C ), then A , B , and C are independent is a FALSE statement. It is also necessary that P ( AB ) = P ( A ) P ( B ), P ( AC ) = P ( A ) P ( C ), and P ( BC ) = P ( B ) P ( C ) hold in order for A , B , and C to be independent events. (b) f X ( u ) is an even function and var ( X ) = 4. F X ( u ) = F X (- u ) for all u,- < u < is a FALSE statement. In fact, F X ( u ) = 1- F X (- u ) for all u,- < u < . E [ X 2 ] = 4 is a TRUE statement. E [ X 2 ] = var ( X ) + ( E [ X ]) 2 = var ( X ) = 4....
View Full Document

Page1 / 4

fall09finalsoln - University of Illinois Fall 2009 ECE 313:...

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