Let $X$ be a $\operatorname{Binomial}(n, p)$ random variable with the probability mass function
$$ \mathbb{P}(X=k)=\left(\begin{array}{l} n \\ k \end{array}\right) p^{k}(1-p)^{n-k}, \quad k=0,1,2, \ldots, n . $$
Let $T$ be a random variable defined as:
$$ T= \begin{cases}1 & \text { if } X=1, \\ 0 & \text { otherwise. }\end{cases} $$
Let $\mathbb{E}(T)$ and $\mathbb{V}(T)$ denote the expectation and the variance of a random variable $T$. Which of the following statements is/are true?
- $\mathbb{E}\left(T^{2}\right)=n p(1-p)^{n-1}$ and $\mathbb{V}(T)=n p(1-p)^{n-1}\left\{\left(1-\left(n p(1-p)^{n-1}\right)\right)\right\}$
- $\mathbb{E}(T)=n p$ and $\mathbb{V}(T)=n p(1-p)$
- $\mathbb{E}(T)=n p(1-p)^{n-1}$ and $\mathbb{E}\left(T^{2}\right)=n p(1-p)^{n-1}$
- $\mathbb{E}(T)=p$ and $\mathbb{V}(T)=p(1-p)$
