Generate an observation from a uniform (0,1) given a density function

Question

Let X have density function $ \begin{cases} f(x) =(\alpha-1)/x^\alpha & \text{x>1} \\ 0 & \text{otherwise} \end{cases}$ , where $\alpha>1$ is a constant.
How would one generate an observation according to X from a uniform (0,1) (random number generator)?

Can anyone clarify what the question is asking for?

Answer 1

There are many procedures for simulating a random variable with given distribution. We will make a guess at which one is intended.

The cumulative distribution function $F_X(x)$ of $X$ is given by $$F_X(x)=\int_1^x (\alpha-1)t^{-\alpha}\,dt=1-\frac{1}{x^{\alpha-1}}$$ (for $x\gt 1)$.

We find the inverse function $G(y)$ of $y=1-\frac{1}{x^{\alpha-1}}$. Solving for $x$ we get $x^{\alpha-1}=\frac{1}{1-y}$, and therefore $$G(y)=\left(\frac{1}{1-y}\right)^{1/(\alpha-1)}.$$ It is a standard fact, probably already proved in your course, that if $U$ is uniform on $(0,1)$, and $F_X^{-1}$ exists, then $F_X^{-1}(U)$ has the required distribution.

Thus to produce an "observation" from our distribution, produce an observation $u$ from the uniform on $(0,1)$. Then our required observation is $\left(\frac{1}{1-u}\right)^{1/(\alpha-1)}$.

Remark: The question in the title and the one in the body are different. We answered the question in the body, since ordinarily we want to simulate a given distribution, given a random number generator that produces pseudo-random observations that have a reasonably close to uniform distribution on $(0,1)$.