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Shkiper50 [21]
3 years ago
15

Given a joint PDF, f subscript X Y end subscript (x comma y )equals c x y comma space 0 less than y less than x less than 4, (1)

(5 pts) Determine the constant c value such that the above joint PDF is valid. (2) (6 pts) Find P (X greater than 2 comma space Y less than 1 )(3) (9 pts) Determine the marginal PDF of X given Y

Mathematics
1 answer:
ioda3 years ago
6 0

(1) Looks like the joint density is

f_{X,Y}(x,y)=\begin{cases}cxy&\text{for }0

In order for this to be a proper density function, integrating it over its support should evaluate to 1. The support is a triangle with vertices at (0, 0), (4, 0), and (4, 4) (see attached shaded region), so the integral is

\displaystyle\int_0^4\int_y^4 cxy\,\mathrm dx\,\mathrm dy=\int_0^4\frac{cy}2(4^2-y^2)=32c=1

\implies\boxed{c=\dfrac1{32}}

(2) The region in which <em>X</em> > 2 and <em>Y</em> < 1 corresponds to a 2x1 rectangle (see second attached shaded region), so the desired probability is

P(X>2,Y

(3) Are you supposed to find the marginal density of <em>X</em>, or the conditional density of <em>X</em> given <em>Y</em>?

In the first case, you simply integrate the joint density with respect to <em>y</em>:

f_X(x)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy=\int_0^x\frac{xy}{32}\,\mathrm dy=\begin{cases}\frac{x^3}{64}&\text{for }0

In the second case, we instead first find the marginal density of <em>Y</em>:

f_Y(y)=\displaystyle\int_y^4\frac{xy}{32}\,\mathrm dx=\begin{cases}\frac{16y-y^3}{64}&\text{for }0

Then use the marginal density to compute the conditional density of <em>X</em> given <em>Y</em>:

f_{X\mid Y}(x\mid y)=\dfrac{f_{X,Y}(x,y)}{f_Y(y)}=\begin{cases}\frac{2xy}{16y-y^3}&\text{for }y

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