Chernoff bound $ \mathbb{P} \left\lbrace \Big| X - \mathbb{E}(X) \Big| > t \right\rbrace $

Question

How to apply the Chernoff bound to upper bound the following probability:

$$ \mathbb{P} \left\lbrace \Big| X - \mathbb{E}(X) \Big| > t \right\rbrace $$

where $X$ follows the distribution given as: \begin{equation*} P(X=0)=0; \end{equation*} \begin{equation*} P(X=m)=\binom{n-1}{m-1} p^{m-1}(1-p)^{n-m} \ \ \mathrm{for} \ \ m=1,...n ; \end{equation*}

This is not exactly a binomial distribution but very similar. The term $\mathbb{E}(X)$ is the expected value of $X$ and $t >0$.

Thank you for your help.

Answer 1

$X=Y+1$ where $Y\sim Binom(n-1,p)$. So If $Y_1, Y_2, \ldots$ are i.i.d. random variables, $Y_i\sim Bernoulli(p)$, then $Y=\sum\limits_{i=1}^{n-1} Y_i$. Apply the Chernoff bound on $Y$. The same estimation holds for $X$, as $X-E(X)\sim Y-E(Y)$.