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Unformatted text preview: UCSD ECE 153 Handout #14 Prof. Young-Han Kim Wednesday, October 29, 2008 Solutions to Old Midterm Exam 1. Light bulbs (20 points). Alice and Bob go shopping for light bulbs. Alice buys two regular light bulbs with life span distributed according to the Exp(1) distribution; she will use these two bulbs one by one. Bob goes for one higher-end bulb with life span distributed according to the Exp(1 / 2) distribution. We will denote X 1 and X 2 for the life spans of Alice’s bulbs and denote Y for that of Bob’s bulb. We assume that X 1 , X 2 , and Y are independent of each other. (a) (5 points) What is the pdf of the total life span of Alice’s bulbs, i.e., the pdf of X 1 + X 2 ? (b) (5 points) Compare the expected life spans of Alice’s choice (two cheap bulbs used sequentially) and Bob’s choice (one expensive bulb). (c) (10 points) What is the probability that Bob’s bulb will outlive Alice’s bulbs? You may find the following facts useful. • The pdf of an Exp( λ ) random variable X is f X ( x ) = braceleftBigg λe- λx , x ≥ , , otherwise. • integraldisplay ∞ xe- λx dx = 1 λ 2 . Solution: (a) Let Z = X 1 + X 2 . Since X 1 and X 2 are independent, f Z ( z ) = ( f X 1 * f X 2 )( z ) = integraldisplay z e- x e- ( z- x ) dx = integraldisplay z e- z dx = ze- z ....
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- Spring '11
- Probability, Probability theory, Alice and Bob, pY |X, pX |Y, Sennari