Question about transformations and sums on uniformly distributed random variables.

Question

I'm looking into a few problems as a hobby of mine, and found myself with the following problem: let $X$ be a random variable uniformly distributed on $[0,1]$.

What is the probability that after $N$ samples $(x_1, x_2, ..., x_n)$ the sum $1/x_1 + 1/x_2 + ... + 1/x_n > K$ for some constant $K$?

Thus far, I've found material focusing on the usefulness of CLT approximations of sampling $X$ and avoiding the difficulties that accompany mass convolution, however I still find that I'm a little out of my element - I only ever had one class on probability. Any pointers would be greatly appreciated.

Answer 1

Depending on the size of $n$ and $k$, the answer may be easy or a mess.

If $X \sim Uniform(0,1)$, then $W = \frac1X$ has pdf $g(w)$:

$$g(w) = \frac{1}{w^2} \quad \quad \text{for } w>1$$

Then, the sum $S_n$ of $n$ such independent random variables $S_n = W_1 + W_2 + ... + W_n$ will have domain of support on $(n,\infty)$. So:

The easy part: For some constant $k$:

$$ \text{ If } k \le n, \quad P(S_n>k) = 1 \quad $$

The messy part occurs otherwise (i.e. if $k > n$):

To illustrate the rapidly increasing computational complexity, here are exact solutions for $n = 1, 2$ and 3:

• If $n = 1$ and $k>1$, then: $\quad P(S_1>k) = \frac{1}{k}$

• If $n = 2$ and $k>2$, then: $\quad P(S_2>k) = \frac{2}{k^2} (k+\log (k-1))$

• If $n = 3$ and $k>3$, then:

$$\quad P(S_3>k) = \frac{1}{(k-1) k^3}\big((k-1) \left(3 k^2-4 i \pi \log (k-1)+\pi ^2\right)+4 (k-1) \text{Li}_2\left(\frac{1}{k-1}\right)-4 (k-1) \text{Li}_2\left(\frac{k-2}{k-1}\right)-4 (k-1) \text{Li}_2(k-1)+2 \log (k-2) (3 (k-2) k+2 (k-1) \log (k-1))\big)$$

where $Li_2$ is the polylog function. So exact solutions get messy quite rapidly :)

---

OP asks:

How did you derive these equations?

It is quite easy. Consider the $n=2$ case. If $X_i \sim Uniform(0,1)$, then, by independence, the joint pdf of $(X_1,X_2)$, say $f(x_1,x_2)$, is:

_{(source: tri.org.au)}

Then, you seek:

_{(source: tri.org.au)}

where I am using the Prob function from the mathStatica package for Mathematica here to automate the nitty-gritties. All done. Same process for $n=3$ etc.

Plot of the pdf of the sum

Here is a quick plot of the pdf of $S_n$, when $n$ = 1, 2 and 3:

_{(source: tri.org.au)}

(As disclosure, I should add that I am one of the developers of the software used above).