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I need to estimate a parameter for Binomially distributed variable.

Suppose I want to estimate the probability of hitting the target. I first do 5 trials out of which I hit the target twice.

Then I try again for another 5 times and this time I hit the target only once.

Now i know how to find the probability estimator for each of these cases separately (I would end up with p=2/5 after the 1st experiment and p=1/5 after the second), but what happens if I have 2 experiments/sets of trials like that. How does this change the procedure? I would still like to find the MLE based on all the data that I have.

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  • $\begingroup$ oh, i meant twice indeed, thanks $\endgroup$
    – LushyIvy
    May 10, 2021 at 21:15
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    $\begingroup$ Are those two experiments independent of each other? $\endgroup$
    – Fiodor1234
    May 10, 2021 at 21:57
  • $\begingroup$ Yes, that is the assumption $\endgroup$
    – LushyIvy
    May 11, 2021 at 8:20

1 Answer 1

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We have the first experiment $Y_{1}\sim Bin(5,p)$ and the second experiment $Y_{2}\sim Bin(5,p)$ and those two experiments are independent.

Then I would like to find the pmf of joint the experiments i.e. the $p(Y_{1},Y_{2})$, however as you stated they are independent so we can rewrite it as $p(Y_{1},Y_{2})=p(Y_{1})p(Y_{2})$ where we know exactly what $p(Y_{i})$ is as $Y_{i}$ is a Binomial random variable.

$$p(Y_{1}=y_{1},Y_{2}=y_{2})=p(Y_{1}=y_{1})p(Y_{2}=y_{2})=\binom{5}{y_{1}}p^{y_{1}}(1-p)^{5-y_{1}}\binom{5}{y_{2}}p^{y_{2}}(1-p)^{5-y_{2}} \\ =\binom{5}{y_{1}} \binom{5}{y_{2}}p^{y_{1}+y_{2}}(1-p)^{5+5-y_{1}-y_{2}} (**)$$

When we use the Maximum Likelihood technique we want to maximize $(**)$ with respect to the parameter of interest, in our case we want to maximize with respect to $p$. So, we will maximize only the part that has $p$, i.e. $L = p^{y_{1}+y_{2}}(1-p)^{5+5-y_{1}-y_{2}}$

Hence, by solving the $\frac{dL}{dp}=0\Rightarrow p = \frac{y_{1}+y_{2}}{5+5}=\frac{2}{10}$

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