The joint probability mass function of the random variables X, Y, Z is p (1,2,3)=p(2,1,1)=p(2,2,1)=p(2,3,2)=1/4 Find E[XY+XZ+YZ].

The joint probability mass function of the random variables X, Y, Z is p$\left(1,2,3\right)=p\left(2,1,1\right)=p\left(2,2,1\right)=p\left(2,3,2\right)=\frac{1}{4}$ Find E[XY+XZ+YZ].
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Haylee Krause
The expected value (or mean) is the sum of the product of each possibility with its probability:
$E\left(XY+XZ+YZ\right)=\sum \left(xy+xz+yz\right)p\left(x,y,z\right)$
$=\left(1\left(2\right)+1\left(3\right)+2\left(3\right)\right)\ast \frac{1}{4}+\left(2\left(1\right)+2\left(1\right)+1\left(1\right)\right)\ast \frac{1}{4}$
$=\left(2\left(2\right)+2\left(1\right)+2\left(1\right)\right)\ast \frac{1}{4}+\left(2\left(3\right)+2\left(2\right)+3\left(2\right)\right)\ast \frac{1}{4}$
$=\frac{11}{4}+\frac{5}{4}+\frac{8}{4}+\frac{16}{4}$
$=\frac{11+5+8+16}{4}$
$=\frac{40}{4}$
=10