A gamma distribution with parameters $,$ has the following density function, where $(t)$ is the gamma function.

$p(x)=\frac{{}^{}}{()}{x}^{-1}{e}^{-x}$

If the posterior distribution is in the same family as the prior distribution, then we say that the prior distribution is the conjugate prior for the likelihood function. Using the Gamma distribution as a prior, show that the Exponential distribution is a conjugate prior of the Gamma distribution. Also, find the maximum a posteriori estimator for the parameter of the Exponential distribution as a function of $$ and $.$