Consider a set of continuous random variablces $Y_1 ... Y_n$, i.i.d, exponentially distributed . with rate parameter $\lambda$.
I showed first that for one single variablce (ie the first) its cumulative distribution. Then I showed that the exponential distribution posesses the so-calles memoryless property. Also I showed that the exp. distribtuion is effectively njust a special case of the gamma distribution.
No my quesion is; how to compute the joint density function (y_1, .;.. y_n) of $Y_1 ... Y_n $ ? ?
And how to assign a gamma-prior with parameters a and b to be the unknown rare parameter \lambda and how to compute the posterior distribution $p(\lambda | Y_1 = y1 ... Y_n = y_n)$ ? , of $\lambda$ given the obvered data ??
Also Im wondering how to compute the posterio expectation value and posterior variance of \lambda....
The density of the distribution of $\lambda$ is proportional to $$\lambda^{a-1}\mathrm e^{-b\lambda}\mathbf 1_{\lambda\gt0},$$ and, conditionally on $\lambda$, the density of each $Y_k$ is proportional to $$\lambda\mathrm e^{-\lambda y}\mathbf 1_{y\gt0},$$ hence the density of the posterior distribution of $\lambda$ conditionally on a sample $(y_k)$ of size $n$ is proportional to $$\lambda^{a-1}\mathrm e^{-b\lambda}\prod_{k=1}^n\left(\lambda\mathrm e^{-\lambda y_k}\right)=\lambda^{a+n-1}\mathrm e^{-(b+y_1+\cdots+y_n)\lambda}.$$ This proves that the posterior distribution of $\lambda$ conditionally on the sample $(y_k)$ is the gamma density with parameters $(a+n,b+y_1+\cdots+y_n).$