# How can you have a density function that depends on a parameter?

Let $X_1,...,X_n$ be a random sample from a distribution with mass/density function $f_X$ that depends on a (possible vector) parameter $\theta$. Then $f_{X_1}(x_1) = f_X(x_1;\theta)$ so that

$f_{X_1,...,X_k}(x_1,...,x_k) = \prod_{i=1}^kf_X(x_i;\theta).$

Could someone please explain what the significance of $\theta$ is in the above definition. I've never seen this before. Is it the mass/density function that depends on the parameter or is it the random sample that depends on the parameter.

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θ is called a parameter. It is an unknown constant that is a part of the formula for the density function. So for example if you have a normal distribution with variance 1 and unknown mean the density would be fx(x1;θ) = [exp(-(x1-θ)$^2$ /2)/√(2π)]. θ is the unknown mean in this case. This is very common in statistics. You will need to get familiar with things like this if you want to be able to follow what is being said on this site about statistics.