Therefore obtained as

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Unformatted text preview: ained as () ∫ ) ( ( ) ) ( ) () ( ) ( () ( ) ( ) ( ) is ) and ) The multiplication is nonzero for 0≤x≤1 and 0≤x-w≤1 therefore () ( { ) ( ) → () { fW(w) 1 -1 0 1 Since z = |w|, (z can take only positive values, the density of negative values of w will be added to absolute equivalent of that value, for example for z = ½ the density will be sum of w = ½ and w = -½ ) then ( ) ( ) will be as follows: f Z ( z) () 2 { 0 1 FZ(z) 1 () { 0 1 c) ∫( ) Alternative Solution: a-b) It is known that the pdf of sum of two independent variables is convolution of pdfs of two random variables(See Sec. 4.1 of textbook). Let W be W = X-Y therefore W can be seen as sum of X and –Y. Therefore fW(w) will be as follows: () ∫ () ( ) The multiplication is nonzero for 0≤x≤1 and 0≤x-z≤1 therefore () { ( ) This gives the same result obtained previously. ( ) ). 3) X ~N(2,1) and Y exponential with parameter 1 ( a) E(X) = 2 (first moment, mean), Var(X) = 1, ( )( ) ( ) ( ( )) E(Y) = 1= , Var(Y) = 1/ ( )( ) =1 () ( ( )) b) ( ) ( ))( (( then E(XY) = 1/2+2*1 = 5/2 ( ))) ( Correlation Coefficient = ( ) (( ) ()() () ( )) = (1/2) c) {( )} ( ) ( ) ( ) For ( ) ( ) ( ) ( ) ( ) For ( ) ( ) ( ) ( Therefore the second estimate has a lower mean square error ) ( ) 4) Given that all random variables are independent [ considered as constant since it is given): [ [∏ [ |] can be written as follows (N can be [ [ [ ∑( ⁄ ) () (⁄) For E[Y], N is no more constant and must be considered as random [[ [( ⁄ ) ] The MGF of the geometric r.v. () similarity between ∑ ( ) and ∑ (⁄) ( )( ∑ () ( () () ) ( ) . Once we notice the ( ), the answer is )=1/6. 5) a) The MGF (Moment Generating Function ) of obtained as follows: () ∫ { } ( which has a uniform distribution on ( ) ∫ { } ( ) is ) b) Given , it is known that the MGF of sum of independent random variables is equal to multiplication of MGF of each random variable in the sum. Therefore the MGF of Z is obtained as follows: () [ [ ( )] ( ) [ ( ) [ ( ( ) () ( ( ) ) c) The first moment of Z is obtained as follows ( )| Using L’Hopital Rule ( ( )| ) ( is obtained as we should since as R1 and R2 are identically distributed. [ [ ) )|...
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