STAT1301 (Assig3)

STAT1301 (Assig3) - 09/10 THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: 09/10 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability & Statistics I Assignment 3 Due Date: October 28, 2009 (Hand in your solutions for Questions 1, 3, 13, 14, 21, 28) 1. Let X be a random variable with probability density function ( ) ( ) ⎩ ⎨ ⎧ < < − − = otherwise 1 1 if 1 2 x x c x f . (a) What is the value of c ? (b) What is the cumulative distribution function of X ? (c) Find and . ( ) X E ( ) X Var (d) Evaluate . ( ) 7 . 3 . ≤ < − X P (e) Find the probability density function of 2 X Y = . 2. The probability density function of X , the lifetime of a certain type of electronic device (measured in hours), is given by ( ) ⎪ ⎩ ⎪ ⎨ ⎧ ≤ > = 10 10 10 2 x x x x f . (a) Find the cumulative distribution function of X . (b) Find . ( ) 15 ≥ X P (c) Find and . ( ) X E ( ) X Var (d) Find ( ) X E . (e) Determine the lower quartile, median, and upper quartile of X . (f) What is the probability that of 6 such types of devices at least 3 will function for at least 15 hours? What assumptions are you making? 3. ( Random number generation ) A general method for simulating a random variable—called the inverse transformation method —is based on the following function : ( ) ( ) { } : min u x F x u F ≥ = where F is a distribution function and 1 < < u . (a) Show that ( ) ( ) x F u x u F ≤ ⇔ ≤ for all ( ) 1 , ∈ u and real x . (b) Use the result in (a), or otherwise, to show that if F is a distribution function and U is a uniform random variable from ( ) 1 , , then ( ) U F X = will be a random variable with distribution function F . (c) Write down the procedure to generate a random variable from ( ) λ Exp . P. 1 09/10 4. A random variable Y is said to follow the double exponential distribution if it has the density function ( ) y e y f λ λ − = 2 1 , ∞ < < ∞ − y where > λ . (a) Find the distribution function of Y . (b) Find the moment generating function of Y . (c) Find and . ( ) Y E ( ) Y Var 5. Prove that for any nonnegative continuous random variable X , ( ) ( ) ( ) ∫ ∞ − = 1 dx x F X E where is the distribution function of X ....
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STAT1301 (Assig3) - 09/10 THE UNIVERSITY OF HONG KONG...

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