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    "# Exercise 6.2: The Olkin-Petkau-Zidek example of MLE fragility\n",
    "\n",
    "<hr>\n",
    "\n",
    "In 1981, [Olkin, Petkau, and Zidek](https://doi.org/10.1080/01621459.1981.10477697) demonstrated an example in which MLE estimates can very wildly with only small changes in the data. We will work through their example in this problem.\n",
    "\n",
    "**a)** Say you are measuring the outcomes of $N$ Bernoulli trials, but you can only measure a positive result; negative results are not detected in your experiment. You do know, however that $N$, while unknown, is the same for all experiments. The number of positive results you get from a set of measurements (sorted for convenience) are *n* = 16, 18, 22, 25, 27. Modeling the generative process with Binomial distribution, $n_i \\sim \\text{Binom}(\\theta, N)\\;\\;\\forall i$, obtain maximum likelihood estimates for $\\theta$ and $N$. *Hint:* You can work out an analytical expression for the MLE of $\\theta$ in terms of $N$, and then you can find $N$ by enumerating $N$.\n",
    "\n",
    "**b)** Now, let's say that the final measurement has 28 positive results instead of 27. Repeat your MLE calculation. How do the results vary?"
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