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A function that increases or decays at a rate proportional to its present value is called an exponential function.

As, X is a random variable that can be used with pdf,

\({{\rm{f}}_{\rm{X}}}{\rm{(x) = }}\left\{ {\begin{array}{*{20}{l}}{{\rm{\lambda }}{{\rm{e}}^{{\rm{ - \lambda x}}}}}&{{\rm{,x}} \ge {\rm{0}}}\\{\rm{0}}&{{\rm{,x < 0}}}\end{array}} \right.\)

With parameter\({\rm{\lambda }}\), it is said to have an exponential distribution.

Denote with

\({{\rm{X}}_{\rm{1}}}\)= time till the first birth,

\({{\rm{X}}_{\rm{2}}}{\rm{ = }}\) time between the first and second births,

.

.

\({{\rm{X}}_{\rm{n}}}{\rm{ = }}\) time between the \({{\rm{n}}^{{\rm{th }}}}\) and \({{\rm{(n - 1)}}^{{\rm{th}}}}\) births,

As, \({{\rm{X}}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,n}}\) are independent and exponentially distributed random variables with appropriate parameters \({\rm{ - }}{{\rm{X}}_{\rm{1}}}\) with parameter \({\rm{\lambda }}\), \({{\rm{X}}_{\rm{2}}}\) with parameter \({\rm{2\lambda }}\),..., \({{\rm{X}}_{\rm{n}}}\) with parameter \({\rm{n\lambda }}\).

Allow joint pdf or pmb for random variables\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\).

\({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{\rm{\theta }}_{\rm{1}}}{\rm{,}}{{\rm{\theta }}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{\theta }}_{\rm{m}}}} \right){\rm{, n,m}} \in {\rm{N}}\)

where\({{\rm{\theta }}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,m}}\)are unknown parameters. The likelihood function is defined as a function of parameters\({{\rm{\theta }}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,m}}\)where function f is a function of parameter. The maximum likelihood estimates (mle's), or values\(\widehat {{{\rm{\theta }}_{\rm{i}}}}\)for which the likelihood function is maximised, are the maximum likelihood estimates,

\({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{{\rm{\hat \theta }}}_{\rm{1}}}{\rm{,}}{{{\rm{\hat \theta }}}_{\rm{2}}}{\rm{, \ldots ,}}{{{\rm{\hat \theta }}}_{\rm{m}}}} \right) \ge {\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{\rm{\theta }}_{\rm{1}}}{\rm{,}}{{\rm{\theta }}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{\theta }}_{\rm{m}}}} \right)\)

As, \({\rm{i = 1,2, \ldots ,m}}\) for every \({{\rm{\theta }}_{\rm{i}}}\). Maximum likelihood estimators are derived by replacing \({{\rm{X}}_{\rm{i}}}\) with \({{\rm{x}}_{\rm{i}}}\).

As, \({{\rm{X}}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,n}}\) are independent and exponentially distributed random variables with appropriate parameters \({\rm{ - }}{{\rm{X}}_{\rm{1}}}\) with parameter \({\rm{\lambda }}\), \({{\rm{X}}_{\rm{2}}}\)

Because of the independence, the likelihood function becomes,

\(\begin{array}{c}{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{x}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;\lambda }}} \right){\rm{ = \lambda }}{{\rm{e}}^{{\rm{ - \lambda }}{{\rm{x}}_{\rm{1}}}}}{\rm{ \times (2\lambda )}}{{\rm{e}}^{{\rm{ - 2\lambda }}{{\rm{x}}_{\rm{2}}}}}{\rm{ \times \ldots \times (n\lambda )}}{{\rm{e}}^{{\rm{ - n\lambda }}\left( {{{\rm{x}}_{\rm{n}}}{\rm{ - \theta }}} \right)}}\\{\rm{ = n \times (n - 1) \times \ldots \times 1 \times }}{{\rm{\lambda }}^{\rm{n}}}{\rm{ \times exp}}\left\{ {{\rm{\lambda }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}} \right\}\\{\rm{ = n! \times }}{{\rm{\lambda }}^{\rm{n}}}{\rm{ \times exp}}\left\{ {{\rm{\lambda }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}} \right\}\end{array}\)

Look at the log likelihood function to determine the maximum.

\(\begin{array}{c}{\rm{lnf}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{x}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;\lambda ,}}} \right){\rm{ = ln}}\left( {{\rm{n! \times}}{{\rm{\lambda }}^{\rm{n}}}{\rm{ \times exp}}\left\{ {{\rm{\lambda }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}} \right\}} \right)\\{\rm{ = lnn! + nln\lambda - \lambda }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}\end{array}\)

The maximum likelihood estimator is generated by taking the derivative of the log likelihood function in regard to \({\rm{\lambda }}\) and equating it to \({\rm{0}}\).

As a result, the derivative,

\(\begin{array}{c}\frac{{\rm{d}}}{{{\rm{d\theta }}}}{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{x}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;\lambda }}} \right){\rm{ = }}\frac{{\rm{d}}}{{{\rm{d\lambda }}}}\left( {{\rm{lnn! + nln\lambda - \lambda }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}} \right)\\{\rm{ = }}\frac{{\rm{n}}}{{\rm{\lambda }}}{\rm{ - }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}\end{array}\)

As a result, solving equation provides the maximum likelihood estimator\(\widehat {\rm{\lambda }}\).

\(\begin{array}{c}\frac{{\rm{n}}}{{\rm{\lambda }}}{\rm{ - }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}{\rm{ = 0}}\\\frac{{\rm{n}}}{{\rm{\lambda }}}{\rm{ = }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}\end{array}\)

For\({\rm{\hat \lambda }}\). Hence, the maximum likelihood estimator of parameter\({\rm{\lambda }}\)is,

\({\rm{\hat \lambda = }}\frac{{\rm{n}}}{{\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{X}}_{\rm{i}}}}}\)

We need the values x, which are the times between successive births, based on the serial birth times. The following formula is used to calculate the values:

\({{\rm{x}}_{\rm{1}}}{\rm{ = 25}}{\rm{.2}}\) the time of the first birth;

\({{\rm{x}}_{\rm{2}}}{\rm{ = 41}}{\rm{.7 - 25}}{\rm{.2 = 16}}{\rm{.5}}\) the period of time between the first and second births;

\({{\rm{x}}_{\rm{3}}}{\rm{ = 51}}{\rm{.2 - 41}}{\rm{.7 = 9}}{\rm{.5}}\) the time between the second and third births;

\({{\rm{x}}_{\rm{4}}}{\rm{ = 55}}{\rm{.5 - 51 = 4}}{\rm{.3}}\) the time between the third and fourth births;

\({{\rm{x}}_{\rm{5}}}{\rm{ = 59}}{\rm{.5 - 55}}{\rm{.5 = 4}}\) the time between the fourth and fifth births;

\({{\rm{x}}_{\rm{6}}}{\rm{ = 61}}{\rm{.8 - 59}}{\rm{.5 = 2}}{\rm{.3}}\) the time between the fifth and sixth births;

As a result, the total can now be calculated as follows:

\(\begin{array}{c}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}{\rm{ = 1 \times }}{{\rm{x}}_{\rm{1}}}{\rm{ + 2 \times }}{{\rm{x}}_{\rm{2}}}{\rm{ + \ldots + 6 \times }}{{\rm{x}}_{\rm{6}}}\\{\rm{ = 1 \times 25}}{\rm{.2 + 2 \times 16}}{\rm{.5 + \ldots + 6 \times 2}}{\rm{.3}}\\{\rm{ = 137}}{\rm{.7}}\end{array}\)

Last but not least, the maximum likelihood estimates of\({\rm{\lambda }}\)is,

\(\begin{array}{c}{\rm{\hat \lambda = }}\frac{{\rm{n}}}{{\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {\rm{i}} {{\rm{x}}_{\rm{i}}}}}\\{\rm{ = }}\frac{{\rm{6}}}{{{\rm{137}}{\rm{.7}}}}\\{\rm{ = 0}}{\rm{.0436}}\end{array}\)

Therefore, \({\rm{\hat \lambda = 0}}{\rm{.0436}}\).