Solution: Problem1-46
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Started by: siromasiroma
On: 1222692955|%e %b %Y, %H:%M %Z|agohover
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Solution: Problem1-46
siromasiroma 1222692955|%e %b %Y, %H:%M %Z|agohover
(1)
W(X,Y)=X+Y
Expected value of W:
(2)
\begin{tabular}{ccl} E(W) & \stackrel{\text{by def.}}{=} & \iint{(x+y)f(x,y)dxdy \ & = & \iint{xf(x,y)dxdy} + \iint{yf(x,y)dxdy \ & \stackrel{\text{indep.} }{=} & \iint{xf_1(x)f_2(y)dxdy + \iint{yf_1(x)f_2(y)dxdy\ & = &\int{xf_1(x)dx}\int{f_2(y)dy} + \int{yf_2(y)dy}\int{f_1(x)dx}\ & = &\int{xf_1(x)dx} + \int{yf_2(y)dy}\ & = & E(X)+ E(Y) \end{tabular}
Variance of W:
(3)
\begin{tabular}{ccl} Var(W) & \stackrel{\text{by def.}}{=} & \iint{((x+y)-E(X+Y)})^2}f(x,y)dxdy \ & = & \iint{((x-E(X))+(y-E(Y)))^2f(x,y)dxdy} \ & = & \iint{(x-E(X))^2f(x,y)dxdy + \\ & & 2\iint{(x-E(X))(y-E(Y))}f(x,y)dxdy + \ & & \iint{(y-E(Y))f(x,y)dxdy\ & \stackrel{\text{indep.} }{=} & \int{(x-E(X))^2f_1(x)dx\int{f_2(y)dy} +\ & & 2\int{(x-E(x))f_1(x)dx}\int{(y-E(y))f_2(y)dy}+\\ & & \int{(y-E(Y))^2f_2(y)dy\int{f_1(x)dx}\ & = & Var(X) + Var(Y) \end{tabular}
last edited on 1222695807|%e %b %Y, %H:%M %Z|agohover by siroma + show more
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