Here is a bivariate function with the variables $i,j$ is given as: $f(i,j) = Wij(i+j),$ $0<i<1,$ $0<j<1.$ Where $W$ is just a constant. Assume that $M=\min(I,J)$ and $N = \max(I,J).$ Compute Correlation$ (M,N).$
From $f(i,j)= \int_0^1\int_0^1 W ij (i+j) dxdy = 1,$ $W = 18/5.$ so that $$f_{AB}(i,j)= \frac{18}{5} ij (i+j) di dj, \ 0 < i, j < 1.$$ $$f (i)= \int_0^1 \frac{18}{5} i j (i+j) dj= \frac{18i}{5}\left(\frac{i}{2}+\frac{1}{3}\right), \ 0 < i < 1$$ $$f (j)= \int_0^1 \frac{18}{5} i j (i+j) di= \frac{18j}{5}\left(\frac{j}{2}+\frac{1}{3}\right), \ 0 < j < 1$$ $$E(I) = \int_0^1 i \frac{18i}{5}\left(\frac{i}{2}+\frac{1}{3}\right) di = \frac{17}{20}= E(J)$$ $$E(I^2)= \int_0^1 i^2 \cdot \frac{18i}{5}\left(\frac{i}{2}+\frac{1}{3}\right) di = \frac{33}{50} = E(J^2)$$ I am stuck beginning here. However, the idea is to find the distribution of $M$, $$1-F_M(m) = P(I> m)P(J> m) = \int_m^i \frac{18t}{5}\left(\frac{t}{2}+\frac{1}{3}\right) dt \int_m^j \frac{18t}{5}\left(\frac{t}{2}+\frac{1}{3}\right) dt,$$ find expectation and variance of $M$. Then by symmetry, the Var$(M)=$ Var$(N):$ $$Var(I)= E(I^2) - (E(I))^2 =Var(J),$$ so that the covariance and correlation between $M$ and $N$ can be computed using: $$Cov(QR) = E(I)E(J) - E(M)E(N)$$ $$\frac{Cov(QR)}{\sqrt{Var(M)Var(N)}}$$ If there is any better way of solving this problem, please let me know. Any help for computing the correlation in this question is most welcome thanks in advance!