Contents
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.
a)What is the mean and the variance of the process Bob uses? Solve algebraically first, then solve numerically.
b)What effect does increasing r to .99 have on the variance?
Solution:
Part A
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^2r^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} $
$ +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} $
$ +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^2r^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} $
$ +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 -6388 widgets$ ^2 $.
Part B
Since the formula for the variance is given by
$ VAR=r\frac{-(n+1)r^n(1-r)+(1-r^{n+1})}{(1-r)^2} $
$ +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 $
Inserting 100 for n and .99 for r, reduces the magnitude of the variance so that the variance is equal to widgets$ ^2 $.
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