homework3 - 1000 cavelry men in the French army one person...

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1 EL 630: Homework 3 1. A fair coin is tossed 4 times. The random variable X is defined as the total number of heads. Write the events { } { } { } { } . 5 ; 2 ; 2 ; 3 = < = X X X X 2. Do the following functions define probability distribution functions? (i) < - = - . 0 , 0 , 0 , 0 , 1 ) ( x x e x F x X α α (ii) < < = . 2 / 1 , 1 , 2 / 1 0 , , 0 , 0 ) ( x x x x x F X (iii) ) ( tan 1 2 1 ) ( 1 x x F X - + = π for . +∞ < < - x 3. In the following cases determine if ) ( x f X is a valid probability density function (p.d.f) for some suitable K . If so find K . If not give reasons. (i) < < - + = . , 0 , 5 5 ), ( sin 1 ) ( 10 otherwise x Kx x f X (ii) (Discrete case) ( 29 . , 2 , 1 , 0 , 1 0 , = < < = = k p Kp k X P k (iii) = - - . , 0 , 0 , ) ( 2 / ) (ln 2 otherwise x e x K x f x X μ Here μ is a known constant and ln stands for natural log. 4. Past statistics (collected over the last 100 years or more) show that out of every
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Unformatted text preview: 1000 cavelry men in the French army, one person has died of horse kick every year. Assuming that this phenomenon has an (approximate) Poisson distribution (this is indeed found to be true!) find the probability that of the present 10000 caverly men not more than 3 will have such an ending. 5. If the random variables X and Y are such that ( 29 ( 29 ξ Y X ≤ for every , then show that ) ( ) ( ϖ Y X F F ≥ for every . +∞ ≤ ≤ ∞-...
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  • Spring '11
  • voltz
  • Probability distribution, probability density function, Cumulative distribution function, probability distribution functions, valid probability density

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