# Binomial MLE based on 2 experiments

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.

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

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 $$\frac{dL}{dp}=0\Rightarrow p = \frac{y_{1}+y_{2}}{5+5}=\frac{2}{10}$$