Let's consider variables $X_1,X_2,..,X_5 \sim exp(1)$ , $Y=1_{[3,+\infty)}(X_1)$ and $T=\sum_{i=1}^5X_i$ .
I want to calculate $E[Y|T=5]$
My attempts:
we know that $T \sim Erlang(5,1)$ (because $T$ is a sum of five independent exponentially distributed random variables with parameter $1$).
$$E[Y|T=5]=\frac{Y1_{\{T=5\}}}{P(\{T=5\})}=\frac{E[1_{[3,+\infty)}(X_1)\cdot 1_{\{T=5\}}]}{P(T=5)}=\frac{E[1_{[3,+\infty)\cap{\{T=5\}}}]}{P(\{T=5\})}=\frac{P([3,+\infty) \cap \{T=5\})}{P(T=5)}=P(X_1 \in [3, + \infty) | \{T=5\})$$
And here i stacked. Can you please have a look at above and tell me if it's correct? Also I have intuitive problem following : Erlang distribution is a continuous distribution so $P(T=5)=0$ if I'm right.
As you have noticed $P(T=5)=0$ and hence the conditional expectation on that set is not well-defined.