Joint Density/Distribution

Question

Let X and Y be jointly continuous random variables with a joint density given by

fXY (x, y) = c where 20 < x < y < 40,

where c is a constant. Find c and the joint distribution function for X and Y .

Answer 1

For this question, we need to integrate the joint density function by the full bounds of x and y and set the result equal to 1. It honestly doesn't matter which variable you start with; just make sure the bounds for the first are either x < y < 40 or 20 < x < y.

∫₂₀⁴⁰ ∫₂₀^y c dx dy

∫₂₀⁴⁰ c(y-20) dy

c(y²/2 - 20y) |₂₀⁴⁰

c(800 - 800 - 200 + 400)

200c = 1

c = ¹/₂₀₀

Thus, for this to be a joint density function, c = ¹/₂₀₀ = 0.005.