Quinlan7g
2022-09-25
Answered

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

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Haylee Krause

Answered 2022-09-26
Author has **10** answers

The expected value (or mean) is the sum of the product of each possibility with its probability:

$$E(XY+XZ+YZ)=\sum (xy+xz+yz)p(x,y,z)$$

$$=(1(2)+1(3)+2(3))\ast \frac{1}{4}+(2(1)+2(1)+1(1))\ast \frac{1}{4}$$

$$=(2(2)+2(1)+2(1))\ast \frac{1}{4}+(2(3)+2(2)+3(2))\ast \frac{1}{4}$$

$$=\frac{11}{4}+\frac{5}{4}+\frac{8}{4}+\frac{16}{4}$$

$$=\frac{11+5+8+16}{4}$$

$$=\frac{40}{4}$$

=10

$$E(XY+XZ+YZ)=\sum (xy+xz+yz)p(x,y,z)$$

$$=(1(2)+1(3)+2(3))\ast \frac{1}{4}+(2(1)+2(1)+1(1))\ast \frac{1}{4}$$

$$=(2(2)+2(1)+2(1))\ast \frac{1}{4}+(2(3)+2(2)+3(2))\ast \frac{1}{4}$$

$$=\frac{11}{4}+\frac{5}{4}+\frac{8}{4}+\frac{16}{4}$$

$$=\frac{11+5+8+16}{4}$$

$$=\frac{40}{4}$$

=10

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