$N$ is a Poisson distribution with mean $4$. Find $\operatorname{Var}(N\mid N \geq 4)$ You are given that $N$ has Poisson distribution with mean $4$. Find $\operatorname{Var}(N\mid N \geq 4)$
I tried to use the definition of variance, where $\operatorname{Var}(X) = E(X^2) - E(X)^2$
Then
$
P(N\mid N \geq4) = 1-P(N <4) = 1 - \dfrac{71}{3}e^{-4}$ 
I'm stuck now about how to use the formula above to calculation the conditional expected value.
Any help would be appreciated.
 A: Let $$M = N \mid N \ge 4.$$  That is to say, $$\Pr[M = k] = \frac{\Pr[N = k]}{\Pr[N \ge 4]}, \quad k = 4, 5, \ldots.$$  Then $$\operatorname{Var}[M] = \operatorname{E}[M^2] - \operatorname{E}[M]^2 = \sum_{k=4}^\infty k^2 \frac{\Pr[N = k]}{\Pr[N \ge 4]} - \left(\sum_{k=4}^\infty k \frac{\Pr[N = k]}{\Pr[N \ge 4]} \right)^2.$$  This is just one way to do the calculation; it may not necessarily be the most elegant.
A: $\newcommand{\var}{\operatorname{var}}\newcommand{\E}{\operatorname{E}}$Let $Y=0\text{ or }1$ according as $N<4$ or $N\ge 4$. For the Poisson distribution the expectation is the same as the variance, so we have
$$
4 = \var(N) = \var(\E(N\mid Y)) + \E(\var(N\mid Y)).
$$
Next
$$
4 = \E(N) = \E(N\mid N < 4)\Pr(N < 4) + \E(N\mid N \ge 4)\Pr(N\ge 4). \tag 1
$$
It is not hard to find $\Pr(N<4)$, and in order to find $\E(N\mid N<4)$ you need four conditional probabilities:
\begin{align}
\Pr(N=0\mid N<4) & = \frac{e^{-4}}{e^{-4} + 4e^{-4} + \frac{16e^{-4}} 2 + \frac{64 e^{-4}} 6} = \frac 1 {1+4+\frac{16} 2 + \frac{64} 6} = \frac 3{71} \\[8pt]
\Pr(N=1 \mid N<4) & = \frac{12}{71} \\[8pt]
\Pr(N=2 \mid N<4) & = \frac{24}{71} \\[8pt]
\Pr(N=3 \mid N<4) & = \frac{32}{71}
\end{align}
So $\E(N\mid N<3) = 156/71 \approx 2.197\ldots$
$\Pr(N<4)$ is readily found and $\Pr(N\ge 4)$ is $1$ minus that.
This makes it possible to find $\E(N\mid N\ge 4)$.
Then we have the random variable $\E(N\mid Y)$:
$$
\E(N\mid Y) = \begin{cases} \E(N\mid N<4) & \text{with the probability $p$ found above}, \\ \E(N\mid N\ge 4) & \text{with probablity }1-p. \end{cases}
$$
Hence
$$
\var(E(N\mid Y)) = \Big( \E(N\mid N<4) - \E(N\mid N\ge 4) \Big)^2 p(1-p).
$$
Proceed similarly to find the other term in $(1)$.
