Consider the random variables $N, H$, and $T$ sampled as follows. First, $N$ is sampled from the Poisson$(10)$ distribution. This means that for each integer $n \geq 0$,
\[
\operatorname{Pr}[N=n]=\frac{e^{-10} 10^{n}}{n !} \text {. }
\]
Then $N$ independent fair coins are tossed. $H$ is then the number of heads, and $T$ the number of tails, obtained in this process. Choose the correct statement from the ones given below.
($\textbf{Note:}$ Recall that for any real number $x, e^{x}$ is defined by the infinite series
\[
e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !},
\]
where $0! =1$ by convention$.)$
- $H$ and $T$ are independent and the variance of $H$ is equal to the variance of $N$.
- $H$ and $T$ are independent and the variance of $H$ is half the variance of $N$.
- $H$ and $T$ are not independent, but they are independent conditioned on $N$.
- $H$ and $T$ are not independent, and $E[H T]<E[H] \times E[T]$.
- $H$ and $T$ are not independent, and $E[H T]>E[H] \times E[T]$.
