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Consider the following joint probability density function of the random variables X and Y:
f(x,y)= {6y 0 o otherwise}
(1) Find the marginal distribution of X and erify that it is a valid density function.
(2) What is the probability that proportion Y is less than 0.5, given that X is 0.7.

Respuesta :

Answer:

Step-by-step explanation:

Given that a joint probability density function of the random variables X and Y: is

[tex]f(x,y) = 6y, 0<y<x<1\\ = ootherwise\\[/tex]

To find the marginal distribution of x

1) To find marginal distribution of x, we integrate with respect to y the region of y.

Here region of y is 0<y<x

So pdf of X = [tex]\int\limits^x_0 {6y} \, dy \\=3y^2_0^x\\=3x^2, 0<x<1[/tex]

2) P(y<0.5/x=0.7)

X is having a continuous pdf and hence P(x=0.7) =0

(For any continuous distribution at a particular point prob is 0)

So we find that P(y<0.5/x=0.7) does not exist.

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