 So in this example, for finding minimal sufficient statistics, we're going to be looking at the beta distribution as defined here. So we recall that as a consequence of Fisher's factorization theorem, we're really looking at the ratio of this in terms of x to this in terms of y. And we're looking for when is it constant with respect to our parameters alpha and beta. So we need to think about recollecting n observations of this. But we'll start with a single observation f of x over f of y. And that's equal to x to the alpha minus 1, 1 minus x to the beta minus 1, divided by y to the alpha minus 1, 1 minus y to the beta minus 1. And then we will have gamma to the alpha, gamma function of alpha plus beta over gamma of alpha, gamma of beta. And then for the denominator bit, it's going to be just invert that. So you can see immediately, you're going to get these parts are going to cancel out. So we're really interested in anything that directly depends on x or y. Now, we're going to look at this in terms of instead of a single observation, we're going to look at it in terms of n observations. And that's going to be the product of all this, i equals 1 to n xi to the alpha minus 1. And we'll say the product i equals 1 to n 1 minus xi to the beta minus 1, divided by the product i equals 1 to n yi alpha minus 1, times the product i equals 1 to n 1 minus yi to the beta minus 1. And then we look at what's in terms of to the power of alpha minus 1, and what's in terms of to the power of beta minus 1. We group those terms together. So we get the product i equals 1 to n xi, divided by the product i equals 1 to n yi to the power of alpha minus 1. And the product i equals 1 to n of 1 minus xi, divided by the product i equals 1 to n 1 minus yi, and all of this is to the power of beta minus 1. So this ratio here is constant with respect to alpha, and only if the product of xi is equal to the product of yi. And it's constant with respect to beta, if and only if the product of 1 minus xi is equal to the product of 1 minus yi. So our minimal sufficient statistics for alpha beta here, is the product of i equals 1 to n xi, and the product i equals 1 to n of 1 minus xi. And so there we found the minimal sufficient statistics for the parameters of beta distribution.