StatORJune06 - probability 1-p . Let X denote the lifetime...

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Part B: Probability Common Qualifying Exam Questions for June 2006 DO 3 out of 4 1. Independent trials are performed. If the i th such trial results in a success with probability i P , compute the expected number and variance of the number of successes that occur in the first n trials. 2. Type i light bulbs function for a random amount of time having a distribution with mean i μ and standard deviation , i ± 2 , 1 = i . A light bulb randomly chosen from a bin of bulbs is a type 1 bulb with probability p , and a type 2 bulb with
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Unformatted text preview: probability 1-p . Let X denote the lifetime of the bulb. Find ) ( X E and ) var( X . 3. Let 4 3 2 1 and , , , X X X X be independent continuous random variables having a common distribution function F and density function f , and let }. { 4 3 2 1 X X X X P I < < < = Compute I . 4. The random variable X is stochastically larger than Y , written , Y X st if for all t , }. { } { t Y P t X P > > Show that if , Y X st where X and Y are nonnegative random variables, then ]. [ ] [ Y E X E...
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