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Question: Let $X_1,\dots,X_n$ be iid exponential rate $\lambda$. Suppose we don't know the observed values of our experiments, but we know that $k$ values were $\le M$ and the remaining $n-k$ were $>M$ for some constant $M$. Find the MLE of $\lambda$.

I need to find the likelihood function $L(\lambda; \vec x) $, which is typically defined as $ \prod_{i=1}^n f(x_i;\lambda)$ for iid data. Now, I intuitively believe that the $L(\lambda; \vec x)$ will equal ${n \choose k} [P(X \le M)]^k [P(X > M)]^{n-k}$, but I'm not satisfied with my explanation why.

The likelihood function is defined as $L(\theta) = f(x_1,\dots,x_n;\theta)$, where $\theta$ is allowed to vary. I don't know how to relate this definition in terms of the pdf to the fact that precisely $k$ observed values were $\le M$. Can someone help with a more rigorous explanation?

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    Your definition of the likelihood for this problem is correct. For discrete models, the likelihood is just the probability that you will see the result of the experiment, as a function of the parameter $\lambda$. – 2017-01-11
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    @user321210 Thanks that is reassuring. The random variables are continuous though, – 2017-01-11
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    The rvs may be continuous, but the data you're modeling is not. You have finitely many observations. – 2017-01-11
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    So you have an estimate of quantile $k/n$. What is the _CDF_ of $X?$ And does that tell you anything about $\lambda?$ – 2017-01-11

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