2

I saw this question in one of my textbooks:

Let U be a random variable having a uniform(0,1) distribution. Describe how to simulate the outcome of the roll of a die using U.

I know that the outcome of rolling a die follows discrete uniform distribution with 6 outcomes, and each outcome has a probability of 1/6. I'm not quite sure how to approach this question. The straightforward solution I thought about was adding 6 unif[0, 1] distributions together to obtain the result of rolling a die. Would that be reasonable? If not, could someone correct me? Thanks!

Paul726
  • 23
  • 3
    Adding six uniform random variables will give you something where values close to $3$ are more likely than others, so not what you want. Instead think about multiplication and rounding – Henry Jan 11 '23 at 18:56
  • 2
    Hint, To simulate flipping a coin, think about how the value of your uniform choice from $[0,1]$ compares to $1/2$. – Ethan Bolker Jan 11 '23 at 18:59

1 Answers1

1

The 6 different sides of the die are all equally likely with probability $\frac{1}{6}$. So if you are given another random variable, just partition its image set into six equally likely subsets.

In your case: If $X$ follows a uniform distribution on $[0,1]$, just define a function $F : [0,1]\rightarrow \{1,...,6\}$ via $$F(x) = 1, \text{ if } x \in\left[0,\frac{1}{6}\right],$$ $$\vdots$$ $$F(x) = 6, \text{if } x \in \left[\frac{5}{6},1\right],$$ and set $Y = F(X)$ as your new variable. This random variable will yield a number from $1$ to $6$ with probability of $\frac{1}{6}$ for each one, i.e. simulate the throw of a die.

Nuke_Gunray
  • 2,816