# Solved – the intuitive meaning behind plugging a random variable into its own pdf or cdf

A pdf is usually written as \$f(x|theta)\$, where the lowercase \$x\$ is treated as a realization or outcome of the random variable \$X\$ which has that pdf. Similarly, a cdf is written as \$F_X(x)\$, which has the meaning \$P(X<x)\$. However, in some circumstances, such as the definition of the score function and this derivation that the cdf is uniformly distributed, it appears that the random variable \$X\$ is being plugged into its own pdf/cdf; by doing so, we get a new random variable \$Y=f(X|theta)\$ or \$Z=F_X(X)\$. I don't think we can call this a pdf or cdf anymore since it is now a random variable itself, and in the latter case, the "interpretation" \$F_X(X)=P(X<X)\$ seems like nonsense to me.

Additionally, in the latter case above, I am not sure I understand the statement "the cdf of a random variable follows a uniform distribution". The cdf is a function, not a random variable, and therefore doesn't have a distribution. Rather, what has a uniform distribution is the random variable transformed using the function that represents its own cdf, but I don't see why this transformation is meaningful. The same goes for the score function, where we are plugging a random variable into the function that represents its own log-likelihood.

I have been wracking my brain for weeks trying to come up an intuitive meaning behind these transformations, but I am stuck. Any insight would be greatly appreciated!

Contents