09_12_2 - Suppose that X and Y are independent the p.d.f of...

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STAT 410 Examples for 09/12/2011 (2) Fall 2011 1. 2.1.6 Let f ( x , y ) = e x y , 0 < x < , 0 < y < , zero elsewhere, be the pdf of X and Y . Then if Z = X + Y , compute P ( Z 0 ) , P ( Z 6 ) , and, more generally, P ( Z z ) , for 0 < z < . What is the pdf of Z ? 1. (continued) Let f ( x , y ) = e x y , 0 < x < , 0 < y < , zero elsewhere, be the pdf of X and Y . Determine the probability distribution of the following random variables: b) W = 2 X + Y ; c) V = X Y ; d) U = Y X X + ; e) T = X Y . 2. When a person applies for citizenship in Neverland, first he/she must wait X years for an interview, and then Y more years for the oath ceremony. Thus the total wait is W = X + Y years. Suppose that
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Unformatted text preview: Suppose that X and Y are independent, the p.d.f. of X is f X ( x ) = 2 / x 3 , x &amp;gt; 1, zero otherwise, and Y has a Uniform distribution on interval ( 0, 1 ). Find the p.d.f. of W, f W ( w ) = f X + Y ( w ). Hint: Consider two cases: 1 &amp;lt; w &amp;lt; 2 and w &amp;gt; 2. 2. (continued) Suppose that X and Y are independent, the p.d.f. of X is f X ( x ) = 2 / x 3 , x &amp;gt; 1, zero otherwise, and Y has a Uniform distribution on interval ( 0, 1 ). b) Let V = X Y. Find the p.d.f. of V, f V ( v ) = f X Y ( v ). Hint: Consider two cases: 0 &amp;lt; v &amp;lt; 1 and v &amp;gt; 1. c) Let U = Y / X . Find the c.d.f. and p.d.f. of U....
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