If an option pays $6 one-quarter of the time and loses $6 three-quarters of the time,then the variance $\sigma$² = : A) 0. B) -3. C) 9. D) 12. E) 27.

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Quizplus · Accepted Answer

The variance of a random variable is calculated using the formula $\sigma^2 = E[X^2] - (E[X])^2$, where $E[X]$ is the expected value of $X$ and $E[X^2]$ is the expected value of $X^2$.First, calculate the expected value ($E[X]$):$$E[X] = (0.25 \times 6) + (0.75 \times -6) = 1.5 - 4.5 = -3$$Then, calculate $E[X^2]$:$$E[X^2] = (0.25 \times 6^2) + (0.75 \times (-6)^2) = (0.25 \times 36) + (0.75 \times 36) = 9 + 27 = 36$$Finally, calculate the variance ($\sigma^2$):$$\sigma^2 = E[X^2] - (E[X])^2 = 36 - (-3)^2 = 36 - 9 = 27$$

If an Option Pays $6 One-Quarter of the Time and Loses