Suppose $X_1,X_2,\ldots,X_n$ is a random sample from a Poisson distribution with mean $λ$. How can I find the conditional expectation $E \left( X_1\times X_2\times X_3 \mid \sum_{i=1}^n X_i= z \right)$?
And $X_1,X_2,\ldots,X_n$ are independent identically distributed Random variables.
My thought:
$E \left( X_1\times X_2\times X_3 \mid \sum_{i=1}^n X_i=z \right)$
$= E \left( X_1 \mid\sum_{i=1}^n X_i=z \right)\times E \left( X_2 \mid\sum_{i=1}^n X_i=z \right)\times E \left( X_3 \mid\sum_{i=1}^n X_i=z \right)$
$= \frac{z}{n}\times \frac{z}{n}\times\frac{z}{n} $