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The Midterm of Probability and Statistics Time:120 mins, April 19, 2007, 2:30-4:30pm and Total 110 points. 1. (10%) In a collection of 12 light bulbs, 3 are defective. The bulbs are selected at random and tested, one at a time, until the third defective bulb is found. Compute the probability that the third defective bulb is the tenth bulb tested. 2. (10%) The mean and standard deviation in midterm tests of a probability course are 72 and 12, respectively. These quantities for final tests are 68 and 15. What final grade is comparable to a midterm grade of 82? 3. (10%) Let X be a discrete random variable with the set of possible values { 1 , 2 , 3 , · · ·} . Suppose that for all positive integers n and m , we have P ( X > n + m | X > m ) = P ( X > n ) . Show that X is a geometric random variable. 4. (10%) Let X be a random variable, g a density function with respect to integration, and φ a differentiable strictly increasing function on ( -∞ , ). Suppose that P ( X x ) = φ ( x ) -∞ g ( z ) dz, -∞ < x < Show that the random variable Y = φ ( X ) has density g .

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