Question

The joint density of X and Y is given by

f (x,y) = xe^(-x(y+1)); x>0,y>0
0; otherwise

Answer 1

\[x e^{-x(y+1)}\]

Answer 2

Find the distribution of Z=XY using distribution function method

Answer 3

@Zarkon

Answer 4

@JamesJ I know you're doing ajp's problem, but I wanna know how to do this one as well if you get the chance

Answer 5

Just compute \[P(Z<z)=P(XY<z)\]

integrate over the region in which \(XY<z\) is true

Answer 6

How to integrate it when it has two variables?

Answer 7

x y < z
x < z/y
You should compute the integral below to F(z) the CDF and F'(z) would be the pdf

\[
\int _0^{\infty }\int
_0^{\frac{z}{y}}x e^{-x
y-x}dxdy
\]
Do you know how to compute this integral?

Answer 8

you compute first the integral below by parts
\[
\int_0^{\frac{z}{y}} x e^{-x
y-x} \, dx= \frac{1-e^{-\frac{(y+1) z}{y}}
\left(\frac{z}{y}+z+1\right)
}{(y+1)^2}
\]
It is messy, who said life is easy.

Answer 9

Then you integrate
\[
\int_0^{\infty }
\frac{1-e^{-\frac{(y+1)
z}{y}}
\left(\frac{z}{y}+z+1\right)
}{(y+1)^2} \, dy= 1- e^{-z}
\]
I have to confess that I did not do the above integral by hand and I refuse to do it by hand.

Answer 10

So
\[
F(z) = 1 - e^{-z}\\
f(z)=F'(z) = e^{-z}
\]

Answer 11

Thank you for your big help. I will try @eliassaab :)

Answer 12

A few friends are having a disagreement about spotters. Select the statement that shows the individual understands the role of the spotter correctly.

Answer 13

The regular price of a bag is $41.45. During a sale, the bag was marked 20% off.

Answer 14

Celina measured the height of a window frame as 5.55 feet, but the actual height was 6 feet.

Answer 15

If 3/4 ton of pebbles covers 4/5 of a garden, how many tons of pebbles are requir

Answer 16

If a sprinkler waters 1/8 of a lawn in 1/4 hour, how much time will it take to wa

Answer 17

The price of oranges at a farm is $1.16 per pound. Which equation can be used to determine c, the total price of n pounds of oranges? (1 point)

Select one:

Answer 18

(04.03 LC)

The table below shows the proportional relationship between the weigh

Answer 19

(04.02 LC)
The table shows the price for different numbers of ice cream cones:

Number of Ice Cream Cones Price (in dollars)
2 4
4 8
5 10
8 16
Does this tab

Answer 20

(04.02 LC)
The table shows the price for different numbers of ice cream cones:

Number of Ice Cream Cones Price (in dollars)
2
4
4
8
5
10
8
16
Does this ta

Answer 21

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Choice A: 6 ounces of raisins for $1.

Answer 22

"A dazzling skyrocket illuminating the drab literary landscape" in sentence 3 is

Answer 23

"A dazzling skyrocket illuminating the drab literary landscape" in sentence 3 is

Answer 24

The graph below shows the relationship between pounds of dog food and total cost, in dollars, for the dog food.

Answer 25

6.03B Muir Graphic Organizer B

Answer 26

The diagram below shows a food web in a woodland.

A diagram of a food web is shown.

Answer 27

Kristen cut her hand on a piece of glass. As the blood clots, how do white blood

Answer 28

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Erosion takes place along the inner edge as water flows relatively sl

Answer 29

After reading about how the Constitution was written and ratified, why it has lasted so long, and some things citizens might do to support it, you get to wr

Answer 30

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Answer 31

13. Four cards are drawn from a standard deck. No cards are returned.
a) Which is more likely:
Event A: drawing a straight (four consecutive cards e.