1

Let $X_1,\cdots,X_{10}$ be i.i.d. random variables, where $X_i$ is from Gamma$(\alpha,\beta)$ with $\alpha>0$ and $\beta>0$. I want to find $E(X_1|\sum_{i=1}^{10} X_i=10)$.

Here's my attempt: Since $X_i$ are i.i.d., $E(X_1|\sum_{i=1}^{10} X_i=10)=E(X_2|\sum_{i=1}^{10} X_i=10)=\cdots=E(X_{10}|\sum_{i=1}^{10} X_i=10)$. Thus, we can find $$\sum_{i=1}^{10}E(X_i|\sum_{i=1}^{10} X_i=10)=E(\sum_{i=1}^{10}X_i|\sum_{i=1}^{10} X_i=10)$$ then divide by 10. However, given $\sum_{i=1}^{10} X_i=10$, the expectation of $\sum_{i=1}^{10}X_i$ should be 10. Thus, $E(X_1|\sum_{i=1}^{10} X_i=10)$ should be 1.

Am I wrong?

Accept Cookies
  • 109
  • 4
  • 2
    The question is nonsensible, but your approach is correct in that the same argument shows $E(X_1\mid \sum X_i)=\sum X_i/n$ – Andrew Oct 18 '23 at 21:45

0 Answers0