Continuous joint random variable

[docnet]

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)
$$P(X\geq 0.5|Y\leq 0.5)=\frac{3b}{16}+\frac{3b^3}{12\cdot 8}$$

 

[LastScattered1090]

Hi @docnet -- I don't think this thread belongs in precalculus math. It clearly involves integration, and anything related to Probability Theory would probably go in Calculus & Beyond, I'd imagine.

You seem to have arrived at a full answer for every sub-part of your question. I think part of the homework template calls for describing what's tripping you up. How can we be of help here?

 

[docnet]

Hi @docnet -- I don't think this thread belongs in precalculus math. It clearly involves integration, and anything related to Probability Theory would probably go in Calculus & Beyond, I'd imagine.

You seem to have arrived at a full answer for every sub-part of your question. I think part of the homework template calls for describing what's tripping you up. How can we be of help here?

I'm not sure.. I was hoping that I made a mistake somewhere so someone could correct me.