Realizations of random variable

Question

Can some of you help me with following exercise?

Provide a procedure to generate realizations of a random variable with CDF (cumulative distribution function) $Fx(x)$ given by:

$Fx(x)=(x+1)/4$ if $-1<x<0$

$Fx(x)=(x+3)/4$ if $0<=x<1$

What is the meaning of generate realizations of random variable?

Thank all!

Answer 1

I think what is meant is: produce (pseudo-) random numbers. Note that for a distribution function $F$ and its inverse $F^{-1}$ and a uniform random variable $U$ it holds that $ F^{-1}(U) $ has distribution $F$. You an google random number generation and find this procedure. In you case:

1. find the inverse of $F$
2. Generate uniform random variables
3. Apply $F^{-1}$

Then you are done. Which programming language do you use?

Edit: As remarked by Macro there is an atom at $0$ with probabilty $\frac12$. I think we can still use the inverse in the following way:

1. find the inverse for $x \in [-1,0)$ and $x \in (0,1]$
2. Generate a uniform $U$

3. Call the generated rv $X$

   3.a) If $U \in [0,\frac14]$ then apply $F^{-1}$ as definded on the left of $0$ and set $X = F^{-1}(U)$

   3.b) If $U \in (\frac14,\frac34)$ then set $X = 0$

   3.c) If $U \in [\frac34,1]$ then apply $F^{-1}$ as definded on the right of $0$ and set $X = F^{-1}(U)$.

This should take care for the atom. Note that I did not care about open or closed intervals in the definition above as the probability for the uniform to reach the the point at the interval end is zero.