# Is the product of uniformly distributed numbers, uniformly distributed too?

My question is simple, I think.

If we took two random natural numbers $a$ and $b$ uniformly distributed in a specific range $[c,d]$, is $ab$ a uniformly distributed too?

What if $a$ and $b$ are not natural numbers, but real numbers?

What if $a$ is a natural number and $b$ a real number?

• The product will lie in a range as well. $[c^2, d^2]$ in case of natural numbers. Aug 11 '14 at 21:03
• By "random" and "true random" do you mean uniformly random?
– Seth
Aug 11 '14 at 21:06
• If they are natural numbers then the products will NOT be uniformly distributed! Aug 11 '14 at 21:12
• One very easy way of seeing this in the discrete case: if your ranges are, e.g., $a\in [2, 10]$ and $b\in [2, 10]$ then the product $ab$ must be in the range $[4, 100]$ - but there's no way of getting e.g. $5\in [4, 100]$ as a product of $ab$. (And a counting version: there are only $9\times 9=81$ results you can get for $ab$, but $97$ numbers in the range $[4,100]$, so you know there must be 'missed' numbers even without specifically knowing about primes.) Aug 11 '14 at 22:18
• @String Let a=c+d-b and d=c+1. Then P(ab=cd)=1 which is technically a uniform distribution. It is the only one I can think of. There may be others but I doubt it. Aug 12 '14 at 1:09

NOTE: This analysis only considers positive real numbers $c,d$ and independent random variables $a,b$ both uniformly distributed in $[c,d]$.

In this case $ab$ is NOT uniformly distributed in $[c^2,d^2]$. Note that $(a,b)$ will be uniformly distributed in $S=[c,d]\times[c,d]$ whereas $ab=k\in[c^2,d^2]$ is a hyperbola that passes through the point $(\sqrt k,\sqrt k)$ in $S$.

Analysis: The probability of $ab<k$ for $k<cd$ will be defined by the equation \begin{align} A\cdot P(ab<k)&=\int_c^{k/c}\left(\frac{k}{x}-c\right)\ dx\\ &=k\ln(k/c^2)-c(k/c-c)\\ &=k\ln(k/c^2)-k+c^2 \end{align} where $A=(d-c)^2$ is the area of the region $S$ and the integral above calculates the area that lies beneath the hyperbola $yx=k\implies y=\dfrac{k}{x}$ yet in the region $S$. So the probabilty does not depend linearly on $k$ which means that the distribution cannot be uniform on $[c^2,d^2]$.

Note that for $k=cd$ the hyperbola passes through the cornes $(c,d)$ and $(d,c)$ of the region. So for $k>cd$ more is cut off due to the boundaries of the region and thus $P(ab<k)$ starts growing slower.

Analysing the case $[c,d]=[1,2]$ entirely leads to the distribution function $$P(ab<k)= \begin{cases} k\ln(k)-k+1&k\in[1,2]\\ k-k\ln(k/4)-3&k\in[2,4] \end{cases}$$ which is illustrated here

where the cases where $k>cd=2$ are covered by calculating the purple area as $$P(ab>k)=\int_{k/2}^2\left(2-\frac{k}{x}\right)\ dx=k\ln(k/4)-k+4$$ and therefore $P(ab<k)=1-P(ab>k)=k-k\ln(k/4)-3$.

If you treat each of them as random variables, then the product is also a random variable. To see this note that if $X_1, \ldots, X_n$ are random variables, then $(X_1, \ldots, X_n)$ is a random vector on $\Bbb R^n$, hence applyi,g the continuous, and therefore measurable function

$$(X_1,\ldots, X_n)\mapsto X_1\ldots X_n$$

We get the product to be the composition of measurable functions, hence measurable. I.e. The product is also a random variable.

Note that the distribution may very well not be the same as before, but whether or not it counts as a random variable is not a question of the distribution type. Indeed, the other answers cover pretty well that it will not be uniform in general, but the power of this approach is that it does not depend on the choice of probability measure for the exact space, so it does not matter if you are talking about the naturals or the reals or something else entirely, nor does it matter what way you choose to,define "randomness."

• The comments from the OP below the question suggests that the real question was about being uniform rather than just random. The original question surely did not ask of that, I agree! Aug 11 '14 at 23:13

Here's a simple example: Suppose $X$ and $Y$ are independent and identically distributed uniform random variables in $[0,1]$. Let $Z = XY$. Clearly, we must have $0 \le Z \le 1$. Thus, if $Z$ were also uniform, we would have for instance $\Pr[Z > 1/2] = 1/2$. But $XY \le X$ and $XY \le Y$, hence $XY > 1/2$ only if $X > 1/2$ and $Y > 1/2$; i.e., $$\Pr[XY > 1/2] < \Pr[(X > 1/2) \cap (Y > 1/2)] = (1/2)^2 = 1/4.$$

No. Say you are choosing between 1 and 4. If you take every possible result you get:

1*1 = 1
1*2 = 2;
1*3 = 3;
1*4 = 4;
2*2 = 4;
2*3 = 6;
2*4 = 8;
3*3 = 9;
3*4 = 12;


The result 4 is more likely appear than any other number and if the input is truly random as sample size increases it will become uniform.

This answer is trivially no for natural numbers.  Let $c=1$ and $d=2$.
Now $P(ab=1)=\frac14$, $P(ab=2)=\frac12$, $P(ab=3)=0$, and $P(ab=4)=\frac14$.

• You are assuming a and b are independent. Let a=2-b. a and b are still both random variables, uniform(1,2). But P(ab=2)=1. ab is distributed uniform(2,2). Aug 12 '14 at 1:01
• (1) I believe that you mean $a=3-b$; but, aside from that, good point. (2) However, I think all the other answers assume independence also. (At least heropup states the assumption explicitly.) Aug 12 '14 at 1:06