# integer is chosen from set$\{1,2,…,9,10\}$. Find $E(X)$ and $Var(X)$.

$X$ is the number of different positive integers that divide the chosen number without remainder. For example, if $6$ is chosen , then the value of $X$ is $4$ since $1,2,3,6$ each divide $6$ without remainder.

First construct a table with two rows listing integers between 1 and 10 and the divisors of each integer (which I have already done).

Second use this table to construct another take with two rows listing the unique values $X$ assumes (i.e listing the range of $X$) and the probability $p(x) = P(X=x)$ for each value.

Next, use your answer from the first to parts of find $E(X)$ and $Var(X)$.

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Are you seeking help or giving us an assignment? –  jay-sun Jan 26 '13 at 19:18

From your first table, you see 1 once, 2 four times, 3 three times, and 4 twice (for a total of 10, obviously). Hence:

P($X = 1) = \frac{1}{10}$,

P($X = 2) = \frac{4}{10}$,

P($X = 3) = \frac{3}{10}$,

P($X = 4) = \frac{2}{10}$.

From this, the expectation is

E[$X$] = $\frac{1}{10} 1 + \frac{4}{10} 2 + \frac{3}{10} 3 + \frac{2}{10} 4 = \frac{26}{10}$,

and the variance follows similarly if you plug it into the formula you surely have (it is $\frac{84}{100}$).

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Thank you so much that is very helpful! –  user59633 Jan 27 '13 at 20:14