A number is chosen at random from the integers 1, 2, 3,...,10. Let [math]X[/math] be the number chosen. Determine [math]Var(X)[/math].

References

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

Solution: A

Say a number is chosen at random from the integers [math]1,\ldots,n[/math] and denote the outcome by [math]X[/math]. Then

[[math]] E[X] = \frac{1}{n}\sum_{i=1}^n i = \frac{n(n+1)}{2n} = \frac{n+1}{2} [[/math]]

and

[[math]] E[X^2] = \frac{1}{n}\sum_{i=1}^n i^2 = \frac{(n+1)(2n+1)}{6n}. [[/math]]

Hence the variance equals

[[math]] \frac{(n+1)(2n+1)}{6n} - \left(\frac{n+1}{2} \right)^2. [[/math]]

Setting [math]n=10 [/math] gives 8.25.