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homework5

# homework5 - y f Y(b X ∼ 2 σ N X is a normal r.v with...

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1 EL 630: Homework 5 Note: Midterm Exam (Oct 25, 1999) 1. The p.d.f of a continuous r.v X is given by < < - = . , 0 , 2 3 , 5 / 1 ) ( elsewhere x x f X Find (a) ( 29 1 2 X P (b) ( 29 . 0 ) sin( X P π 2. The probability of heads p of a random variable is itself a random variable X (i.e., X = p ). Further X is uniform in the interval (0,1). (a) Find the probability that at the next tossing the coin will show tails. (b) The coin is tossed 10 times and heads showed 6 times (observation). Find the a posteriori probability that X is between 0.35 and 0.75 (i.e., we need ), | 75 . 0 35 . 0 ( B X P < < where B is the given observation). 3. Let X have p.d.f ). ( ) ( x U e x f x X α α - = Compute the p.d.f of (a) . 3 X Y = (b) . 3 2 + = X Y 4. Let X be a discrete random variable with probability mass function ( p.m.f ) ( 29 . , , 2 , 1 , 0 , n k q p k n k X P k n k = = = - ( X is a Binomial r.v). Find the p.m.f of (a) 1 2 + = X Y (b) X Y = (c) . X e Y - = 5. (a) Let X be a positive r.v of the continuous type with p.d.f ). ( x f X Find the p.d.f of . 1 X X Y + = If, in particular, X is uniformly distributed in (0,1), find

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Unformatted text preview: ). ( y f Y (b) X ∼ ) , ( 2 σ N ( X is a normal r.v with parameters 0 and 2 ). (i) Define . 2 X Y = Find ). ( y F Y (ii) Define . | | X Z = Find ). ( z f Z 2 6. (a) Let X be the r.v with p.d.f ∞ < < ≤ < ≤ = . 1 , 2 / 1 , 1 , 2 / 1 , , ) ( 2 x x x x x f X Find the p.d.f of the r.v . / 1 X Z = (b) X is given to be uniform in ( 29 . , π-Define . sec X Y = Compute ). ( y f Y 7. (a) Define the event M as ( 29 ( 29 . 2 1 ≤ < ∪ ≤ = X X M Find ) | ( M x F X and ) | ( M x f X in terms of ) ( x F X and ) ( x f X where X is a nonzero r.v over the entire real axis. (b) Let X be a r.v with p.d.f given by < < = . , , , / 2 ) ( 2 elsewhere x x x f X Define . sin X Y = Find the p.d.f of the r.v Y ....
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homework5 - y f Y(b X ∼ 2 σ N X is a normal r.v with...

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