# Find a distribution function on a probability space with singleton set range

Let $X$ be a real-valued variable defined on a complete probability space $\left(\Omega,F,P\right)$ with singleton set range $R(X)={x}$ for some $x\in R$.

I would like to be able to do the following:

i) find a distribution function $F_{X}$ of $X$.
ii) find a probability density function $f_{X}$ of $X$ (if it exists)
iii) Draw sketches of $F_{X}$ and $f_{X}$ with some justifications
iv) draw some conclusions from i),ii) and iii) if possible.

I'm a little bit confused as to where to begin. Hints and suggestions would be very much appreciated. Thanks.

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i) $F_X(t)=0$ if $t<x$ and $F_X(t)=1$ if $t\ge x$. ii) $f_X$ does not exist. iii) Please do. iv) ?
Thanks. I'll do iii. :) . Please why doesn't $f_{X}$ exist. –  Godwin May 12 '11 at 17:49