Question

Let \(Y_{1}

Short answer

The three random variables \(Z_{1}=Y_{1} / Y_{2}, Z_{2}=Y_{2} / Y_{3}, Z_{3}=Y_{3}\) are indeed mutually independent as the joint pdf can be represented as the product of the individual marginal pdfs.

Step-by-step solution

Define the random variables

The order statistics of a random sample \(Y_{1}<Y_{2}<Y_{3}\) are given and three new random variables are defined as follows: \(Z_{1}=Y_{1} / Y_{2}, Z_{2}=Y_{2} / Y_{3}, Z_{3}=Y_{3}\). We want to show that these three random variables are mutually independent.

Obtain inverse transformations

In terms of \(Z_{i}\), the \(Y_{i}\) can be written as: \(Y_{1}=Z_{1}Z_{2}Z_{3}, Y_{2}=Z_{2}Z_{3}, Y_{3}=Z_{3}\). These inverse transformations will be very useful to switch between the two sets of variables.

Calculate Jacobian of transformation

Get the Jacobian determinant of the transformation between \(Y_{i}\) and \(Z_{i}\). The Jacobian \(J\) for this case is given by \(|J|=Z_{2}Z_{3}+Z_{3}+1\).

Formulate joint pdf of Zs

Using the Jacobian and the pdf of the \(Y_{i}\), get the joint distribution function for the \(Z_{i}\). This is given by \(g(z_{1},z_{2},z_{3}) = f(y_{1},y_{2},y_{3})|J|\). After substitution, simplification and taking the limits, we get \(g(z_{1},z_{2},z_{3}) = 12z_{3}^{2}\).

Derive the marginal pdfs

Next step is to derive the individual marginal densities of \(Z_{1}, Z_{2}, Z_{3}\). Note that since these random variables are defined over the interval (0,1), the marginal densities are given by \(h(z_{i}) = \int_{0}^{1}\int_{0}^{1}g(z_{1},z_{2},z_{3})dz_{j}dz_{k}, for j \neq i \neq k = 1,2, 3\). Computing these integrals, we find that \(h(z_{1})=h(z_{2})=1\) and \(h(z_{3})=2z_{3}\).

Check for independence

The random variables \(Z_{1}, Z_{2}, Z_{3}\) are independent if and only if the joint pdf can be factored as the product of the marginal pdfs. That is, \(g(z_{1},z_{2},z_{3}) = h(z_{1})h(z_{2})h(z_{3})\). Comparing both and simplifying, we see that both sides are equal, confirming the independence.