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- Homework Statement
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- Relevant Equations
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(a)
$$\int_0^1\int_0^1x+cy^2 dxdy=\int_0^1 [\frac{x^2}{2}+cxy^2]_0^1dy= \int_0^1\frac{1}{2}+cy^2 dy=[\frac{y}{2}+\frac{cy^3}{3}]_0^1=\frac{1}{2}+\frac{c}{3}=1$$
$$\Rightarrow c=\frac{3}{2}$$
(b) The marginal pdf of X is
$$f_X(a)=\int_0^1 f_{X,Y}(a,b)db=\int_0^1 x+\frac{3}{2}y^2 dy=[xy+\frac{3y^3}{6}]_0^1=x+\frac{1}{2}$$The marginal pdf of Y is
$$f_Y(b)=\int_0^1 f_{X,Y}(a,b)da=\int_0^1 x+\frac{3}{2}y^2 dx=[\frac{x^2}{2}+\frac{3xy^2}{2}]_0^1=\frac{1}{2}+\frac{3y^2}{2}$$(c) The joint CDF of (X,Y) is
$$\int_0^a\int_0^b(x+cy^2)dxdy=\int_0^y[\frac{x^2}{2}+cxy^2=]_0^ady=\int_0^y\frac{a^2}{2}+cay^2dy=$$
$$\frac{a^2y}{2}+\frac{3ab^3}{6}\Rightarrow F_{X,Y}(a,b)= \frac{a^2b}{2}+\frac{3ab^3}{6}$$
(d)
$$P(X\geq 0.5|Y\leq 0.5)=\frac{3b}{16}+\frac{3b^3}{12\cdot 8}$$
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