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X and y are independent continuous random variables and have the same distribution Fx = 1- (1/t) for t > 1. We define two new variables W and Z, where W = min ( x and Y) and Z = max ( X and Y) Find the mean of w and mean of z. I wanted to understand how can we solve this using jacobian transformation. Here both the variables are in terms of t, so how do we do the transformation and apply the limits.

Sundaresh Subramanian
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  • You will need considerable care with the Jacobian, because neither transformation is everywhere differentiable and the mapping $(X,Y)\to(W,Z)$ is not one-to-one. Perhaps the easiest way to obtain the means is to use the ["Tail probability integral"](https://stats.stackexchange.com/a/222497/919). This obviates any need to compute densities. – whuber Jun 14 '20 at 16:36

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I'm not sure a transformation approach using a Jacobian is the best way to do this.

By independence, for the minimum we have

$$P(W > t) = P(X > t, Y > t) = P(X > t)P(Y>t) = 1/t^2,$$ for $t > 1.$ So $F_W(t) = P(W \le t) = 1 - 1/t^2.$

Also, $$F_Z(t) =P(Z \le t) = P(X \le t, Y \le t) = P(X\le t)P(Y\le t) = (1-1/t)^2,$$ for $ t > 1.$

I will leave it to you to find the density function and the expectation of $W$---and to discuss the expectation of $Z.$ Perhaps see Wikipedia on Pareto distrributions.

BruceET
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