Question

Determine the normalizing constant K in the joint density function

Answer 1

Integrate over the region where \(0<y<x<\infty\) and our total density should amount to \(1\). Thus \(K\) can be found readily.

Answer 2

Like this right?\[K\int\limits_0^\infty {\int\limits_0^x {\frac{1}{y}{e^{ - \frac{{{y^2}}}{2} - \frac{x}{y}}}} } dydx = 1\]. I've been struggling to solve this but with no hope though :(

Answer 3

$$\int_R\frac{K}ye^{-\frac{y^2}2-\frac{x}y}\mathrm{d}A=K\int_0^\infty \mathrm{d}y\int_0^x\mathrm{d}x\frac1ye^{-\frac{y^2}2-\frac{x}y}=K\lim_{c\to\infty}\int_0^c\mathrm{d}y\int_0^x\mathrm{d}x\frac1ye^{-\frac{y^2}2-\frac{x}y}$$