The joint density function for the random variables [math]X,Y [/math] equals

[[math]] f_{X,Y}(x,y) = \begin{cases} cxy^3, y^2 \lt x \lt y, 0 \lt y \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]

for a constant [math]c[/math]. Determine the marginal density of [math]2Y^{1/2}[/math] given [math]X=1/2[/math].

• [[math]] g(z)= \begin{cases} \frac{z^7}{6}, \sqrt{2} \lt z \lt 2^{3/4} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} \frac{64z^3}{3}, \frac{1}{2} \lt z \lt \frac{1}{\sqrt{2}} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} z^3, \sqrt{2} \lt z \lt 2^{3/4} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} \frac{255z^7}{1688}, \frac{1}{2} \lt z \lt \frac{1}{\sqrt{2}} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} \frac{2^{7/2}z^{5/2}}{5}, 0 \lt z \lt 2 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]