# 180ahw6 - MATH 180A Introduction to Probability Homework...

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MATH 180A – Introduction to Probability Homework 6 (due 11/20/06) 1. Consider an urn with 1 red, 2 green and 3 blue balls. 5 balls are drawn from that urn without replacement. Let X be the number of red balls picked and Y the number of green balls picked. Find the joint distribution of X and Y . 2. Consider an urn with 1 red, 2 green and 3 blue balls. 100 balls are drawn from that urn with replacement. What is the probability that: (a) There are 15 red, 35 green and 50 blue balls? (b) All balls are of the same color? (c) There are more blue balls than red and green balls together? 3. Consider X and Y random variables with joint probability density function f ( x, y ) = C ( y - x ) e - y , 0 < x < y, y (0 , ) . (a) Are X and Y independent? (b) Compute C . (c) Compute E ( X ), var( X ), E ( Y ), var( Y ). (d) Compute the marginal densities of X and Y . 4. Same as above with f ( x, y ) =
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Unformatted text preview: f ( x, y ) = C x 2 y (1-y 2 ) , x ∈ (-1 , 1) , y ∈ (0 , 1) . 5. Factory A produces a certain type of resistor. The resistor lifetime has the exponential distribution. The average resistor lasts 1,000 hours. Suppose we buy several resistors from factory A to design a couple of circuits. (a) For circuit 1, we set them in parallel, so that circuit 1 works if any of the resistors work. Compute the minimum number of resistors we need to buy in order for circuit 1 to last at least 10,000 hours with probability 95%. (b) For circuit 2, we set them in series, so that circuit 2 works if all of the resistors work. Compute the maximum number of resistors we can buy in order for circuit 2 to last at least 100 hours with probability 95%....
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