Binomial distribution

Link: Probability Statistical test

Calculate probability of binomial distribution §

$$pr(x|n,p)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x}=\frac{n!}{x!(n-x)!}{p}^x(1-p{)}^{n-x}$$

Binomial test §

When we use a binomial distribution to calculate a P-value, it’s called a binomial test. It means that we would calculate the probability of this scenario, plus any other scenario that are equally rare or rarer.

Maximum likelihood in binomial distribution §

When we want to know the likelihood, we simply swap the location of x and p in above equation on the left and the right side remains the same. In this way, we can change the p while x and n are fixed. e.g.

$$L(p|n,x)=L(p=0.25|n=7,x=4)=\frac{7}{4!(7-4)!}0.25^x(1-0.25{)}^{7-4}=0.058$$

It means the likelihood of p=0.25 given that 4 out of 7 people would randomly prefer orange flavor is 0.058; as a comparison, the likelihood of p=0.5 is 0.273. As such, we can find different likelihood depends on p, and the Maximum likelihood is the peak value.

By doing the math for solve when the derivative is 0, we will get a value for:

$$p=\frac{x}{n}$$

This is the maximum likelihood estimate for p when there’s n trials and x successes, which is also the simple average.