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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.

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  • $\begingroup$ 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. $\endgroup$
    – whuber
    Apr 3, 2014 at 15:40
  • $\begingroup$ 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" ($F_X . F_Y$). $\endgroup$
    – Glen_b
    Apr 4, 2014 at 4:10
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    $\begingroup$ I meant distribution of the product of the random variables. Sorry, I was not sure what to use for product operator. $\endgroup$
    – Roman
    Apr 4, 2014 at 15:17
  • $\begingroup$ 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. $\endgroup$
    – Roman
    Apr 4, 2014 at 15:21
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    $\begingroup$ See stats.stackexchange.com/questions/166737/… $\endgroup$ Dec 9, 2022 at 3:53

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