91Ó°ÊÓ

There are 21 observations are taken at random from an exponential distribution for which the mean μ is unknown and, the average of 20 of these observations is 6,We need to determine the M.LE. of μ if the other observation could not be determined, it was known to be greater than 15.

The likelihood function of the exponential distribution is given by

\(f\left( {x;\mu } \right) = \frac{1}{{{\mu ^{20}}}}{e^{ - \frac{1}{\mu }\sum\limits_{i = 1}^{20} {{x_i}} }} \times {e^{ - \frac{{15}}{\mu }}}\)

The average of 20 observations is 6 , hence the sum is 120 and the likelihood function becomes

\(f\left( {x;\mu } \right) = \frac{1}{{{\mu ^{20}}}}{e^{ - \frac{1}{\mu }\sum\limits_{i = 1}^{20} {{x_i}} }} \times {e^{ - \frac{{135}}{\mu }}}\)

Taking log we get

\(\log f\left( {x;\mu } \right) = - 20\log \mu - \frac{{135}}{\mu }\)

Now taking partial derivative and setting it equal to zero we get

\(\begin{array}{c}\frac{{\partial L\left( \mu \right)}}{{\partial \mu }} = - \frac{{20}}{\mu } + \frac{{135}}{{{\mu ^2}}}\\ = 0\end{array}\)

This indicates

\(\begin{array}{l}20\mu = 135\\ \Rightarrow \mu = 6.75\end{array}\)

Hence the required M.L.E. is 6.75