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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?
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.
