A die is loaded so that the probability of a face coming up is proportional to the number on that face. The die is rolled with outcome [math]X[/math]. Find [math]Var(X)[/math].

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: B

We have

[[math]]P(X=i) = \frac{i}{\sum_{i=1}^6 i } = \frac{i}{21}[[/math]]

,

[[math]] E[X] = \frac{1}{21}\sum_{i=1}^6 i^2 = \frac{91}{21} [[/math]]

and

[[math]] \operatorname{E}[X^2] = \frac{1}{21}\sum_{i=1}^6 i^3 = 21. [[/math]]

Hence the variance of [math]X[/math] equals 21 - (91/21)² = 2.22.