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(B) Let $X = \alpha - \beta$. The possible values of $X$ range from $-5$ to $5$.Since $\alpha$ and $\beta$ are independent and identically distributed

Vedclass · Accepted Answer

$48$

Vedclass · Answer

(B) Let $X = \alpha - \beta$. The possible values of $X$ range from $-5$ to $5$.Since $\alpha$ and $\beta$ are independent and identically distributed discrete uniform variables on ${1, 2, 3, 4, 5, 6}$,the variance of $\alpha$ is $	ext{Var}(\alpha) = \frac{n^2 - 1}{12} = \frac{36 - 1}{12} = \frac{35}{12}$.Similarly,$	ext{Var}(\beta) = \frac{35}{12}$.Since $\alpha$ and $\beta$ are independent,$	ext{Var}(\alpha - \beta) = 	ext{Var}(\alpha) + 	ext{Var}(-\beta) = 	ext{Var}(\alpha) + 	ext{Var}(\beta)$.$	ext{Var}(\alpha - \beta) = \frac{35}{12} + \frac{35}{12} = \frac{70}{12} = \frac{35}{6}$.Here,$p = 35$ and $q = 6$. Since $35$ and $6$ are coprime,we have $p = 35$.The prime factorization of $p$ is $35 = 5^1 	imes 7^1$.The sum of the positive divisors of $p$ is $(5^0 + 5^1)(7^0 + 7^1) = (1 + 5)(1 + 7) = 6 	imes 8 = 48$.

Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha - \beta$ is $\frac{p}{q}$,where $p$ and $q$ are coprime,then the sum of the positive divisors of $p$ is equal to

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