density and $f$

Question

If $Y_1, . . . , Y_{10}$ are independent random variables with a density f and the joint density $f_{Y_2},_{Y_{10}}$ equals $x^6y^2(x^3−y^3)^4$, if $0<x<y<2$ and $f_{Y_2},_{Y_{10}} (x, y) = 0$, how do I go about finding f?

Answer 1

Let the CDF of $X_i$ be $F$. Then$$f_{X_2,X_{10}}(x,y)=\begin{cases}\binom{10}7\binom31F(x)f(x)f(y)[F(y)-F(x)]^7,&x\le y\\0,&\text{otherwise}\end{cases}$$From here it is visible that $f(y)=ky^2,0<y<2$ and $0$ otherwise. Can you find $k$?

Answer 2

The joint probability density function of the second and tenth order statistic will be: $$f_{X_{(2)},X_{(10)}}(x,y)=\frac{10!}{7!}\,F(x)\,f(x)\,\big(F(y)-F(x)\big)^7\,f(y)$$ Where, of course, $F(x)$ is the cumulative distribution function.