Calculate the Variance of $\min(N_k,p)$

Question

I am trying to compute the variance of a random variable $\min(N_k,p)$, where $N_k$ is a random variable and $p$ is a fixed number. I have computed the expectation using the conditional expectation, which is as given below $$\mathbb{E}[\min(N_k, p)] = \mathbb{E}[N_k \mid N_k \leq p] *\mathbb{P}(N_k \leq p)+ \mathbb{E}[\min(N_k, p)\mid N_k > p] * \mathbb{P}(N_k > p).$$ Now, to compute the variance, we need to compute $\mathbb{E}[(\min(N_k,p))^2]$. I don't have any idea how I should proceed to compute this because I don't have access to $\mathbb{P}(\min(N_k,p)=i)$.

Answer 1

You can still apply conditional probability, as $$ \begin{align*} \text{Var}(\min(N_k, p)) &= E[(\min(N_k, p))^2] - E[\min(N_k, p)]^2 \\ &= E(N_k^2 | N_k \leq p)P(N_k \leq p) + p^2 P(N_k > p) - \left[ E(N_k | N_k \leq p) P(N_k \leq p) + p P(N_k > p) \right]^2 \end{align*} $$