Inverse of cumulative distribution function

cumulative-distribution-functionsprobability distributionsrandom variables

Let $F(x)$ is the cumulative distribution function and $P(x)$ is the (given) probability distribution function and $X$ is a random variable.
Can anybody please intuitively explain,

  • Why can the inverse of the CDF give us the random variable $X$?
  • Why can't we find the random variable $X$ from the PDF?
  • I wonder what $F$ and $F^{-1}$ get and return?

Thanks.

Best Answer

What do you mean 'find the rv X from the PDF'? If you know pdf, you 'know' rv (whatever you mean by that). The main property of CDF is that it is a non-decreasing function. Specifically, for continuous rvs, you can find $x_0$, s.t.: $$ p=P(X \leq x_0)=F_{X}(x_0) \Leftrightarrow F^{-1}_{X}(p) = x_0 $$
also referred to as a percentile.

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