Suppose a coin is tossed $n$ times and it is then observed that the numbers of head (the success event) appeared is at least $k<n$. I'm interested in knowing the conditional expectation of no. of heads. If $h$ denotes the no. of heads then what is $E[h|h\geq k]$? For this, we need to know the conditional pmf of $h$. I have two thoughts here:
(i) Conjecturing $\gamma$ as the success probability, calculate the conditional pdf. Here I have the following expression: \begin{equation} E[h|h\geq k]= \bigg(\frac{1}{p(h\geq k)}\bigg) \big[k.p(h=k)+(k+1).p(h=k+1)+ ....+n.p(h=n)\big] \end{equation} The relevant probabilities involved are nothing but binomial probabilities and are calculated using the conjecture $\gamma$. Does this result above have a simpler expression?
(ii) Use MLE method to infer the probability of success first and then use this inferred value to perform (i). Now, I looked hard, I could not find an MLE for this. I know for an observed event $(h=k)$, the MLE is $\hat{\gamma}=k/n$ but couldn't find the same for the event $(h \geq k)$.
Please help.
