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Practice1 - Practice problems for Midterm 1 Problem 1 You...

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Practice problems for Midterm 1 Problem 1 You are given two coins, one fair and the second one that comes up heads with probability . 7. Initially, you have no information which coin is biased (the two coins look the same). You toss each of the two coins twice, and coin A comes up heads more times than coin B does. What is the probability that coin A is the biased one? Problem 2 The pdf of a continuous random variable X is given by f X ( x ) = a if 0 < x < 1 b if - 1 < x < 0 . The random variable has the mean EX = 1 / 4. ( a ) Find a and b and plot the density. ( b ) Compute and plot the cumulative distribution function of X . ( c ) Find P ( X 1 / 2), P ( X = - 2 / 3) and P ( X = 0). Problem 3 A continuous random vector ( X, Y ) has a joint pdf given by f X,Y ( x, y ) = 6 y if 0 < y < 1 and y < x < 2 - y 0 otherwise . ( a ) Find the marginal probability densities of X and Y . Are X and Y independent? ( b ) Compute the probability P ( X + Y > 1). Problem 4 A sack contains several boxes, each containing a die.
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