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180ahw5 - m such that a resistor has the same probability...

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MATH 180A – Introduction to Probability Homework 5 (due 11/06/06) 1. Suppose X has the uniform distribution on (0 , 1). Compute the probability density function and expected value of (a) X α , where α > 0; (b) log( X ); (c) exp( X ); (d) sin(2 πX ). 2. A coin is tossed 10,000 times, and we assume the tosses to be independent of each other. (a) Suppose the coin is fair. Find k N such that, if X is the number of times the coin lands heads in 10,000 tosses, P ( | X - 5000 | ≤ k ) 0 . 95 . (b) Assume the coin landed heads 5,150 times. Would you say that the coin is fair? 3. Factory A produces a certain type of resistor. The resistor lifetime has the exponential distribution. The average resistor lasts 1,000 hours. (a) Compute the probability that it will last more than 1,100 and less than 1,200 hours. (b) Compute the median lifetime of a resistor, meaning find
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Unformatted text preview: m such that a resistor has the same probability of lasting more than m hours as lasting less than m hours. 4. David buys a used radio from a friend who bought it new 3 years ago. The radio is in working order. (a) Assuming the lifetime of a radio is exponentially distributed and lasts an average of 7 years, compute the probability that David will enjoy his radio for at least 5 years. (b) Answer the same question, now assuming the lifetime of a radio has PDF f ( x ) = (4 / 21) x e-(2 / 7) x , x > . (Note: the expected lifetime is also 7 years here.) 5. X has the following probability density function f ( x ) = C 1 + x 2 . ( X is said to have the Cauchy distribution .) Find the probability density function of (a) X 2 (b) | X | (c) p | X | ....
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