# Produce the random variable for an asset from a uniformly distributed random varible

I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). I cannot understand the following question(not the answer, but the question itself):

Question 5.2: Suppose an asset takes values from a discrete set $$v(j)$$ and the probability of $$v(j)$$ is $$p(j)$$. Write an algorithm that produces the random variable for this asset from a uniformly distributed random variable.

What is the meaning of "produces the random variable for this asset from a uniformly distributed random variable", can any expert give an example to show what it means? Really appreciate your help!

The question requires you to provide a method which uses uniform random variables and transforms them to generate realizations of the described asset values.

To give a bit more general answer: this is solved by the inverse transform sampling method. The main idea is to obtain realizations of a random variable $$x$$ with any given distribution function $$F(x)$$, by using random numbers $$u$$ ~ $$U(0,1)$$ and transforming them.

To do this, you need first need to obtain the distribution function $$F(x)$$ (in your case you have a probability mass function to begin with), and calculate the inverse cdf $$F^{-1}(x)$$. Finally, for $$u$$ ~ $$U(0,1)$$, the values $$x = F^{-1}(u)$$ have a distribution $$F(x)$$.

• Thanks a lot for the quick response, detailed explanation! Really appreciate it! Commented Nov 13, 2019 at 17:01

Say your asset can take the discrete values {1,2,3,4} with probabilities {0.4, 0.1, 0.2, 0.3}.

The question is to derive a sampling procedure that returns either {1,2,3,4} with the right probabilities according to the underlying distribution.

The solution is to use a random uniform variable ($$u \sim U(0,1)$$)and allocate it based on the following:

if $$u < 0.4 \implies 1$$
if $$u \geq 0.4, u < 0.5 \implies 2$$
if $$u \geq 0.5, u < 0.7 \implies 3$$
if $$u > 0.7\implies 4$$

• Thanks a lot for the quick response and great example! Really appreciate it! Commented Nov 13, 2019 at 17:01