Question

Stat/Probability

Why is the likelihood function for the Binomial distribution just its pdf? Don't we normally multiply the pdf's?

Answer 1

Like, for a Poisson distribution, we say
\[L(\lambda)=\prod_{i=1}^{n}\frac{e^{-\lambda}\lambda^{x_i}}{x_i !}\]

But for the Binomial, we just write:
\[L(p)=\left(\begin{matrix}n \\ x\end{matrix}\right)p^x(1-p)^{n-x}\]. Why is is not \[L(p)=\prod_{i=1}^{n}\left(\begin{matrix}n \\ x\end{matrix}\right)p^x(1-p)^{n-x}\]

Answer 2

Isn't the likelihood function defined as \(L(\theta)=\prod_{i=1}^{n}f(x_i)\) where f(x) is the pdf of the distribution in question. I'm not quite sure why I don't get it for the Binomial.

Answer 3

Maybe this link helps https://onlinecourses.science.psu.edu/stat504/node/28

Answer 4

doesn't help me and i have read it three times now
it just says what the likelihood function is, not how to get it

Answer 5

the thing is that most of the distributions we used, we multiplied out the pdf's, like I showed for the Poisson above. But for some reason, they never explain why they don't do the same for the binomial o_O

Answer 6

no as far as i can see there is no explanation, just says what it is

Answer 7

Yeah I can't seem to find a site that explains it either. Anyway thanks for your help