Distribution of Web Visits

The distribution of daily visitors among websites follows approximately an exponential distribution.

\[P(X=k)={\lambda}e^{- \lambda k}\]

.
It is a rule of thumb that 20% of the websites gets 80% of the visitors. Assuming there are

\[n\]

, estimate the value of

\[\lambda\]

?
Using the rule of thumb, the upper limit is

\[0.2n\]

and the value of the integral is 0.8

\[P(X \le 0.2n)=0.8=\int^{0.2n}_0 {\lambda}e^{- \lambda x}dx=[-e^{-\lambda x}]^{0.2n}_0\]

\[0.8=-(e^{-0.2 \lambda n} - e^0)=1- e^{- 0.2 \lambda n}\]

(Assuming

\[e^{-\lambda n}\]

equals zero)

\[0.8=1-e^{-0.2 \lambda n}\]

\[0.2=e^{-0.2 \lambda n} \rightarrow \lambda= \frac{ln(0.2)}{-0.2 n}=\frac{5 ln(5)}{n}\]