Mathematics: Suppose that X; Y have constant joint density on the triangle with corners at (4; 0), (0; 4), and the origin. a) Find P(X 3; Y 3). b) Are X and Y independent

First turn 20% into a decimal, (.20) than multiply .20 and 80 which makes 16.00. Subtract 16.00 from 80. The answer is $64.00...

Suppose that X; Y have constant joint density on

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jeanniebyrd54

30.01.2022

Mathematics

The triangle (call it T ) has base and height 4, so its area is 1/2*4*4 = 8. Then the joint density function is

$f_{X,Y}(x,y)=\begin{cases}\frac18&\text{for }(x,y)\in T\\0&\text{otherwise}\end{cases}$

where T is the set

$T=\{(x,y)\mid 0\le x\le4\land0\le y\le4-x\}$

(a) I've attached an image of the integration region.

P(X

(b) X and Y are independent if the joint distribution is equal to the product of their marginal distributions.

Get the marginal distributions of one random variable by integrating the joint density over all values of the other variable:

$f_X(x)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy=\int_0^{4-x}\frac{\mathrm dy}8=\begin{cases}\frac{4-x}8&\text{for }0\le x\le4\\0&\text{otherwise}\end{cases}$

$f_Y(y)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dx=\int_0^{4-y}\frac{\mathrm dx}8=\begin{cases}\frac{4-y}8&\text{for }0\le y\le4\\0&\text{otherwise}\end{cases}$

Clearly, $f_{X,Y}(x,y)\neq f_X(x)f_Y(y)$ , so they are not independent.

Suppose that X; Y have constant joint density on the triangle with corners at (4; 0), (0; 4), and th

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Suppose that X; Y have constant joint density on the triangle with corners at (4; 0), (0; 4), and the origin. a) Find P(X < 3; Y < 3). b) Are X and Y independent

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