# Conditional Density Function on Underlying Exponential

Problem Statement: Suppose $$Y_1, Y_2,\dots, Y_n$$ are a random sample from an exponential distribution with mean $$\theta.$$ Let $$\displaystyle U=\sum_{i=1}^n Y_i.$$ Find the conditional density function for $$Y_1$$ given $$U:\; f_{Y_1|U}(y_1|u).$$

Note: This is essentially Exercise 9.86b in Mathematical Statistics with Applications, 5th Ed., by Wackerly, Mendenhall, and Scheaffer. The problem is posed in the context of $$U$$ as a sufficient statistic, and of using the Rao-Blackwell Theorem to find the MVUE of $$e^{-t/\theta}.$$

My Work So Far: The conditional density function is given by \begin{align*} f_{Y_1|U}(y_1|u) &=\frac{f(y_1,u)}{f_U(u)}. \end{align*} Now the density function for $$Y_1$$ is given by $$f_1(y_1)=\frac{1}{\theta}\,e^{-y_1/\theta},$$ and the sum function $$U=\sum Y_i$$ would then have, by virtue of the mgf, \begin{align*} m_U(t) &=\prod_{i=1}^n(1-\theta t)^{-1}\\ &=(1-\theta t)^{-n}, \end{align*} which is the mgf for a Gamma distribution with parameters $$\beta=\theta$$ and $$\alpha=n.$$ It follows that $$f_U(u)=\frac{u^{n-1}\,e^{-u/\theta}}{\Gamma(n)\,\theta^n}.$$ Now $$u=y_1+y_2+\cdots+y_n,$$ so that $$y_1=u-y_2-y_3-\cdots-y_n.$$ By the logic we have already used, if we let $$S=\sum_{i=2}^nY_i,$$ then the distribution function for $$S$$ is a Gamma with parameters $$\beta=\theta$$ and $$\alpha=n-1.$$ Also, $$u=y_1+s,$$ or $$y_1=u-s.$$ Note that $$y_1$$ and $$s$$ are independent. It follows that \begin{align*} f_{(Y_1,S)}(y_1,s) &=f_{Y_1}(y_1)\,f_S(s)\\ &=\frac1\theta\,e^{-y_1/\theta}\, \frac{s^{n-2}\,e^{-s/\theta}}{\Gamma(n-1)\,\theta^{n-1}}\\ &=\frac{s^{n-2}\,e^{-(y_1+s)/\theta}}{\Gamma(n-1)\,\theta^n}. \end{align*}

My Question: I think I'm on the right track, but I also think I'm missing something fairly straight-forward. How do I push through to get the desired conditional probability density function?

I have examined a few questions, especially this one. I do not understand the method of solution there, and the notation they are using is confusing: are they working with $$U$$ or $$Y_n?$$ This question is taking a different approach (doing the expectation without knowing the distribution function), and so doesn't help me.

• stats.stackexchange.com/questions/252692, which poses the same exercise, presents a different kind of solution. I'm not voting to close, though, because you are really asking about your particular method with an aim to addressing a related problem. You still might find some of the ideas used there--especially concerning the exchangeability of the $Y_i$--will be helpful in general.
– whuber
Jul 14, 2021 at 21:45