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I am having difficulty understanding likelihood functions. If we have a probability density function of a random variable $X$ like this:

$f_{X} (x)=ax^2 + bx + c$ (i.e a simple quadratic polynomial), then how to go about finding the likelihood function for x.

All examples of a likelihood function involve some parameter theta. In this example, what is that theta?

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    $\begingroup$ No such $f$ can be a density function because its integral will diverge (or be zero), no matter what values $a$, $b$, or $c$ may have. You must specify a domain over which it is defined (with it implicitly equal to zero elsewhere). Neglecting this little detail has been the origin of many, many questions about likelihood functions here. I suspect you would benefit from searching our site for highly-voted questions about likelihoods; I know they provide plenty of the kinds of examples you seek. $\endgroup$
    – whuber
    Jun 17, 2015 at 19:21

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In likelihood estimation we look for a function of the likelihood of the data, for various values of theta. So in MLE, the data is given, and theta is a variable. In your example, a, b and c are your theta's.

The likelihood function will be the product of the probability density function f(x) over all x (all the data). Your example will probably not have a solution. But, just for the idea, your likelihood function should represent the likelihood, given the data, for all possible theta values.

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