Show that $T\mid (X_1=0) \sim \text{Binomial}((n-1)m,p)$

Question

Given a binomial distribution $(m,p)$ and $T = \sum_{i=1}^{n} X_i$ is a complete sufficient stat. How can i show that $$T\mid (X_1 =0) \sim \text{Binomial}((n-1)m,p)$$

So far i know that $$T\mid X_1 = \frac{\sum_{i=1}^{n} X_i * (1-P)^{m}}{(1-p)^{m}}$$ but I dont know how to proceed from here. Can i get i hint on how to solve this?

Answer 1

I think you mean this: $X_1, \ldots, X_n$ is a random sample of size $n$ from the binomial distribution with parameters $m$ and $p$, $T = \sum_{i=1}^n X_i$, and you want to know the conditional distribution of $T$ given $X_1 = 0$.

Given $X_1 = 0$, $T = \sum_{i=2}^n X_i$, the sum of $n-1$ independent random variables, each with this same binomial($m, p$) distribution. Such a sum has binomial distribution with parameters $m(n-1)$ and $p$.