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Unformatted text preview: ained as
() ∫ )
( ( )
) (
) () ( ) (
() ( )
(
)
( ) is
) and ) The multiplication is nonzero for 0≤x≤1 and 0≤xw≤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: ab) 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 = XY 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≤xz≤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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 Spring '12
 MübeccelDemirekler
 Electronics Engineering

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