Question

The joint density function for X and Y is given as: f(x, y) = kxy for 0 < x < 2y < 1. Find the value of the constant k for which the p.d.f is legitimate. If the video does not work, click here to go to YouTube directly.

Answer 1

For f(x,y) to be a legitimate PDF, we should have

f(z,y)dyd.r = 1 Jr=0 Jy=

= L S Pojdydr = 1 = * L. - Worlyde 1 Jr=0 Jy= Jr=0 Jy=

1= ;[& 1=fir /ܚ

$\Rightarrow k\int_{x=0}^{1}[\frac{x^2}{8}-\frac{x^4}{8}]dx=1 \Rightarrow k[\frac{x^3}{24}-\frac{x^5}{40}]_{0}^{1}=1 \Rightarrow k[\frac{1^3}{24}-\frac{1^5}{40}]=1 \Rightarrow k=60$