# Solved – Maximum Likelihood estimator for family of binomial distributions

For the below example, I am considering Heads as a success and Tails as a failure, when I toss a coin.

(Ex: The first row in the the below tables says, when I tossed the coin 10 times I got 3 Successes and the probability of success is 0.3).

Binomial Distribution Example Now, considering the fact that the Probability of successes might change by increase in trials, I know the maximum likelihood estimator of binomial distribution is Number of Successes/ Total Number of Trials. I feel calculating the MLE for this kind of data is not that straightforward, COuld someone tell me if I am missing something?

P.S: This is a research based question. Any help would be appreciated. Thanks in advance.

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Since per clarification comment, we are tossing the same coin, then in each single Bernoulli trial the probability is the same, \$p\$, it is not affected by number of trials (assuming also an unbiased way of tossing). If moreover we can assume that all Bernoulli trials are independent, and that each sub-sample consists of different tosses (i.e. the \$n=20\$ sample does not contain the \$10\$ tosses of the \$n=10\$ sample), then, viewed together, we have an independent but non-identically distributed sample of realizations from \$10\$ Binomials that have the same unknown probability parameter, but different "number of trials" parameters (although known and deterministic), \$S_i(n_i,p), i=1,2,3,…,10\$, corresponding to \$n\$-parameters \$10,20,30,…,100\$.

Then the joint likelihood of this sample is (ignoring constants that do not include the unknown parameter)

\$\$L propto prod_{i=1}^{10}p^{k_i}(1-p)^{n_i-k_i} = p^{sum k_i}(1-p)^{sum (n_i-k_i)}\$\$

where \$k_i\$'s are the obtained number of successes

So the log-likelihood is

\$\$ln L = left(sum_{i=1}^{10}k_iright)ln p + left(sum_{i=1}^{10}(n_i-k_i)right)ln (1-p)\$\$

You should get

\$\$hat p = frac {sum_{i=1}^{10}k_i}{sum_{i=1}^{10}n_i}\$\$

as should be expected, since you pooled i.i.d. Bernoulli draws, and so the estimator treated them as \$sum_{i=1}^{10}n_i\$ draws from a Bernoulli \$(p)\$ RV in which we had \$sum_{i=1}^{10}k_i\$ successes.

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