Dude91's Third Bonus Point Problem

Question:

Bob owns a company that produces n=100 widgets each day. The probability that a widget is produced without defect is r=.9. What is the mean and the variance of the process Bob uses?

Solution:

If X is taken to be the number of correctly produced widgets made each day, then the expected value of X is
$ E(X)= \sum_{k=0}^n kr^k $
Since
$ \frac{1-r^(n+1)}{1-r}= \sum_{k=0}^n r^k $
Taking the derivative $ \frac{d}{dr} $ of both sides will yield
$ \frac{-(n+1)r^n(1-r)+(1-r^(n+1))}{(1-r)^2}= \sum_{k=0}^n kr^(k-1) $
Multiply both sides by r to see the form of the expected value in the problem:
$ r\frac{-(n+1)r^n(1-r)+(1-r^(n+1))}{(1-r)^2}= \sum_{k=0}^n kr^(k) $

To find the variance, the formula
$ VAR=E(x^2)-(E(x))^2 $
can be used.
$ E(x^2) $ can be expanded to find that
$ VAR=(\sum_{k=0}^n (k^2)(r^k))-(E(x))^2 $
The formula
$ r\frac{-(n+1)(r^n)(1-r)+(1-r^(n+1))}{(1-r)^2}= \sum_{k=0}^n kr^(k) $
can be used to derive the formula for $ E(x^2) $. To do this, take the derivative $ \frac{d}{dr} $ of both sides to find that
$ \frac{-(n+1)r^n(1-r)+(1-r^(n+1))}{(1-r)^2}<\math><br><math>+r\frac{((1-r)^2)(-n(n+1)r^(n-1)+n(n+1)(r^n))-(-(n+1)(r^n)(1-r)+(1-r^(n+1)))(-2+2r)}{(1-r)^4}= \sum_{k=0}^n (k^2)r^(k-1) $
Multiplying both sides by r yields the expression for $ E(x^2) $ to be
$ r\frac{-(n+1)r^n(1-r)+(1-r^(n+1))}{(1-r)^2}<\math><br><math>+(r^2)\frac{((1-r)^2)(-n(n+1)r^(n-1)+n(n+1)(r^n))-(-(n+1)(r^n)(1-r)+(1-r^(n+1)))(-2+2r)}{(1-r)^4}= \sum_{k=0}^n (k^2)(r^k) $
Therefore, the formula for the variance is given by
$ VAR=r\frac{-(n+1)r^n(1-r)+(1-r^(n+1))}{(1-r)^2}<\math><br><math>+(r^2)\frac{((1-r)^2)(-n(n+1)r^(n-1)+n(n+1)(r^n))-(-(n+1)(r^n)(1-r)+(1-r^(n+1)))(-2+2r)}{(1-r)^4}-(r^2)(\frac{-(n+1)r^n(1-r)+(1-r^(n+1))}{(1-r)^2})^2 $
When 100 is inserted in for n and .9 is inserted in for r, the variance can be found to equal .

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