Midterm1

# Midterm1 - 2 if c-1< x< c c if c< x< 2 c 1 4 if 2...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Midterm 1 The question sheet has two sides. The exam is closed books and notes. You have 90 minutes. All problems have equal weight. Show your work. GOOD LUCK! Problem 1 Two identical boxes contain (crooked) coins. Each coin in box 1 shows heads with probability 1/4, and tails with probability 3/4, while each coin in box 2 shows heads with probability 1/3, and tails with probability 2/3. Twice we choose one box at random and take a coin from that box (the second time we do not remember which box was chosen the ﬁrst time around). Each of the two coins is then tossed once. If the two coins show the same thing (i.e. both show a head or both show a tail), what is the probability that the two coins were taken from the same box? Problem 2 The pdf of a continuous random variable X is given by f X ( x ) = 1 /

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Unformatted text preview: 2 if c-1 < x < c c if c < x < 2 c 1 / 4 if 2 c < x < 2 c + 1 (and equals zero for other values of x ), where c is a positive constant. ( a ) Find c and plot the density. ( b ) Compute and plot the cumulative distribution function of X . ( c ) Find P (0 < X ≤ 3 / 4 or X > 3 / 2), P ( X = 1) and P (the distance from X to the origin is greater than 1). Problem 3 A continuous random vector ( X,Y ) has a joint pdf given by f X,Y ( x,y ) = ± e-y if x-1 < y < x + 1 , y > otherwise . ( a ) Find the marginal probability densities of X and Y . Are X and Y independent? ( b ) Compute the probability P ( X + Y < 1). 1 Problem 4 Let Y 1 and Y 2 be independent standard uniform random variables. That is, each one of them has the density f ( y ) = 1 if 0 < y < 1, and 0 otherwise. Compute E h± max( Y 1 , 2 Y 2 )-min( Y 1 ,Y 2 ) ² 2 i . 2...
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## This note was uploaded on 10/29/2011 for the course MATH 3310 at Cornell.

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Midterm1 - 2 if c-1< x< c c if c< x< 2 c 1 4 if 2...

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