# Probability density of the product of independent identically distributed random variables

Let $$X_1, X_2, \dots , X_n$$ be a random sample from the PDF $$f(x;\theta)=\theta x^{\theta -1},\;\; 0 0$$ What is the probabilty that $$\prod_{i=1}^n X_i > t$$?

The joint pdf is $$f(x_1,\dots,x_n;\theta)=\theta^n \left(\prod_{i=1}^n x_i\right)^{\theta -1},\;\; 0 0$$ So \begin{align}\mathbb{P}\left(\prod_{i=1}^n X_i > t\right) &= \int_{\prod_{i=1}^n x_i > t}f(x_1,\dots,x_n;\theta)dx_1\dots dx_n\\ &= \int_{\prod_{i=1}^n x_i > t\\0 Is evaluating this integral the best way of solving this problem, and if so, what's the fastest way to compute the integral? It seems like we should use the fact that the boundary of the integral is similar to the integrand.

• Cross-posting is against site rules. Apr 13, 2019 at 21:21

2. Since in your problem, $$0\le X \le 1$$, logs are negative, so to get a nonnegative random variable with a density function which can be compared to standard distributions, we study the distribution of $$-\log X = \log X^{-1}$$.
3. Doing that you will find that $$-\log X$$ has an exponential distribution.
5. So the answer will be that $$\prod_i X_i$$ has a log-gamma distribution.