# What are the possible simple ways to estimate $P(XY\geq \text{value})$ where $X,~Y$ are two independent beta variables?

I have two independent variables with distributions: $$X \sim \text{beta}(\alpha_1,\beta_1)$$ and $$Y \sim \text{beta}(\alpha_2,\beta_2).$$

I would like to estimate $$P(X Y \geq \text{val})$$, and I am looking for a simple method, possibly approximation, so that I won't need to work with product of distributions.

What are the possible (simple) approaches to solve this problem? I can probably simplify the question to accepting/rejecting the assumption of whether $$X Y=\text{val}$$ with 95% confidence but I am not sure if it helps.

• For integral values of $\alpha_1,\beta_1,\alpha_2,\beta_2$ this probability is a linear function of $\log(\text{val})$ with coefficients that are polynomials in $\text{val}$. Approximation is not necessary, though, because numerical integration can obtain the values as accurately as you like.
– whuber
Apr 3, 2014 at 15:40
• When you say $X*Y$ and write "product of distributions" do you mean convolution of the densities (which is suggested by "$*$"), or do you mean the distribution of the product of the random variables ($X\,Y$)? Neither would be the "product of distributions" ($F_X . F_Y$). Apr 4, 2014 at 4:10
• I meant distribution of the product of the random variables. Sorry, I was not sure what to use for product operator. Apr 4, 2014 at 15:17
• I understand that simulation method can be used. For example, I could draw k pairs$(x,y)$ based on their distributions, multiple them and count how many of them are less than $val$. I wonder if there are simpler and computationally more efficient methods. α1,β1,α2,β2 are indeed integral. Apr 4, 2014 at 15:21
• Dec 9, 2022 at 3:53