A bet wins 8 dollars with probability 1/5 and loses 2 dollars with probability 4/5. What is the standard deviation of the net payoff?

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A bet wins $8$ dollars with probability $\frac{1}{5}$ and loses $2$ dollars with probability $\frac{4}{5}$ . What is the standard deviation of the net payoff?

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Hint

Check the mean first; it may simplify the variance.

Solution

The mean is $\frac{8}{5} - \frac{8}{5} = 0$ .
So the variance equals the second moment: $\frac{64}{5} + \frac{16}{5} = 16$ .
The standard deviation is the square root: $16 = 4$ .

Answer: $4$

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