A joint density function is given by

[[math]] f(x,y) = \begin{cases} kx, \,\, 0 \lt x \lt 1 \,\, \textrm{and} \,\, 0 \lt y \lt 1 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

where [math]k[/math] is a constant. Calculate [math]\operatorname{Cov}(X,Y)[/math].

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Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Define [math]g(x) = kx [/math] and [math]h(y) = 1 [/math]. Then [math]f(x,y) = g(x) h(x) [/math]. In other words, [math]f(x,y)[/math] can be written as the product of a function of [math]x[/math] alone and a function of [math]y[/math] alone. It follows that [math]X[/math] and [math]Y[/math] are independent. Therefore, [math]\operatorname{Cov}[X, Y] = 0.[/math]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.