# Product of two discrete uniform distributions

Let $$a$$ and $$b$$ be positive integers with $$b > a$$.

Let $$A$$ and $$B$$ be two discrete uniform distributions over the integer intervals $$[[1, a]]$$ and $$[[1, b]]$$ respectively.

Then, given any $$t \in [[1, a b]]$$, what is the value of

$$\Pr(A\cdot B = t )?$$

If this is too hard, can we know at least $$\Pr(A\cdot B \le t )?$$

-- EDIT --

Actually, $$A\cdot B = t$$ means that we sampled an $$\alpha$$ from $$A$$ and a $$\beta$$ from $$B$$ such that $$\alpha \cdot \beta = t$$. So, it means that both $$\alpha$$ and $$\beta$$ are divisors of $$t$$, $$\alpha$$ is smaller than or equal to $$a$$, and $$\beta = t/\alpha \le b$$.

So, let $$Div(x) := \{ y \in \mathbb{N}^* : y \text{ divides } x\}$$ and $$D_{t, a, b} := \{(\alpha, \beta) \in Div(t) \times Div(t) : \alpha \le a \text{ and } t / \alpha \le b\}$$, then, we have

$$\Pr(A\cdot B = t) = \frac{|D_{t, a, b}|}{ab}$$

However, that characterization is not very useful, since computing the divisors of $$a$$ and of $$b$$ is very computationally expensive.

Do you know another way of computing this probability?

• You can make a table. Then it is easy to deduce a pdf. – callculus Sep 13 at 16:42
• But when $a$ and $b$ are big, doing such a table is not feasible. – Hilder Vítor Lima Pereira Sep 14 at 12:12