Let $X_i$ denote the percentage of votes cast in a given election that are for candidate $i$, and suppose $X_1$ and $X_2$ have a joint density function
$f_{X_1,X_2}(x,y)= 3(x+y)$ if $x\geq0;y\geq0;0\leq x+y \leq1$ else $f_{X_1,X_2}(x,y)= 0$.
Find $E[X_i]$ for $i=1,2$?
Is it correct to say $E[X_i] = \int_0^1 \int_0^{1-y} (x+y)f_{X_1,X_2}(x,y)$?
Best Answer
To calculate $E[X_1]$ and $E[X_2]$ respectively you have calculate first the marginal densitiy functions:
$f_{X_1}(x)=\int_0^{1-x} 3(x+y) \, dy=\frac32(1-x^2)$
$f_{X_2}(y)=\int_0^{1-y} 3(x+y) \, dx=\frac32(1-y^2)$
Then
$E[X_1]=\int_0^1 x\cdot f_{X_1}(x) \, dx $
$E[X_2]=\int_0^1 y\cdot f_{X_2}(y) \,dy$