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# Expected Value

If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.

# Expected Value - Independent Variables

Let $$X$$ be a uniform random variable over $$\{0, 1, \dots, 6\},$$ and $$Y$$ a random variable over $$\{0, 1, \dots, 4\}$$ with $$P(Y=y) = \frac{y}{10}.$$ If $$X$$ and $$Y$$ are independent, what is $$E[XY] ?$$

If $$X$$ and $$Y$$ are independent random variables such that $$E[X] = 12$$ and $$E[Y] = 17 ,$$ what is $$E[ (X-4)( Y- 5) ] ?$$

If $$X$$ and $$Y$$ are independent random variables such that $$E[X] = 11,$$ $$E[Y] = 20 ,$$ and $$E[ (X-c)( Y- 9) ] = 88,$$ what is $$c?$$

If $$X$$ and $$Y$$ are independent random variables such that $$E[X] = 15$$ and $$E[ (X-4)( Y- 9) ] = 99,$$ what is $$E[Y] ?$$

If $$X, \ Y, \ Z,$$ and $$W$$ are independent random variables such that $$E[X] = 6,$$ $$E[Y] = 5 ,$$ $$E[Z] = 4 ,$$ and $$E[W] = 11 ,$$ what is $E[ (X-3)( Y- 3)( Z- 2)( W- 3) ] ?$

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