Log likelihood and Maximum likelihood

[cajswn]

I'm not sure how to get this first derivative (mainly where does the 4 come from?)
I know x̄ is the sample mean (which I think is 1/2?)
Can someone suggest where to start with finding the log-likelihood?

I know the mass function of a binomial distribution is:

Thanks!

 

[GJA]

Hi cajswn,

To determine the likelihood function we need to create the joint probability distribution. Since the 4 individual trials of the binomial process are independent, we simply multiply the 4 individual probability density functions to get the needed joint probability distribution: $$L = p^{2}(1-p)^{2}.$$ Now take the logarithm to get the log-likelihood function: $$l = 2\ln p + 2\ln (1-p).$$ Since $\bar{x} = 1/2$ we have $2=4\bar{x}$ and $2 = 4-4\bar{x}$. Using these two identities we get: $$l = 4\bar{x}\ln p +(4-4\bar{x})\ln (1-p).$$ From here, take the derivative of $l$ with respect to $p$ to get the result you're looking for.