1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
OLga [1]
3 years ago
14

If X and Y are independent continuous positive random

Mathematics
1 answer:
Leni [432]3 years ago
6 0

a) Z=\frac XY has CDF

F_Z(z)=P(Z\le z)=P(X\le Yz)=\displaystyle\int_{\mathrm{supp}(Y)}P(X\le yz\mid Y=y)P(Y=y)\,\mathrm dy

F_Z(z)\displaystyle=\int_{\mathrm{supp}(Y)}P(X\le yz)P(Y=y)\,\mathrm dy

where the last equality follows from independence of X,Y. In terms of the distribution and density functions of X,Y, this is

F_Z(z)=\displaystyle\int_{\mathrm{supp}(Y)}F_X(yz)f_Y(y)\,\mathrm dy

Then the density is obtained by differentiating with respect to z,

f_Z(z)=\displaystyle\frac{\mathrm d}{\mathrm dz}\int_{\mathrm{supp}(Y)}F_X(yz)f_Y(y)\,\mathrm dy=\int_{\mathrm{supp}(Y)}yf_X(yz)f_Y(y)\,\mathrm dy

b) Z=XY can be computed in the same way; it has CDF

F_Z(z)=P\left(X\le\dfrac zY\right)=\displaystyle\int_{\mathrm{supp}(Y)}P\left(X\le\frac zy\right)P(Y=y)\,\mathrm dy

F_Z(z)\displaystyle=\int_{\mathrm{supp}(Y)}F_X\left(\frac zy\right)f_Y(y)\,\mathrm dy

Differentiating gives the associated PDF,

f_Z(z)=\displaystyle\int_{\mathrm{supp}(Y)}\frac1yf_X\left(\frac zy\right)f_Y(y)\,\mathrm dy

Assuming X\sim\mathrm{Exp}(\lambda_x) and Y\sim\mathrm{Exp}(\lambda_y), we have

f_{Z=\frac XY}(z)=\displaystyle\int_0^\infty y(\lambda_xe^{-\lambda_xyz})(\lambda_ye^{\lambda_yz})\,\mathrm dy

\implies f_{Z=\frac XY}(z)=\begin{cases}\frac{\lambda_x\lambda_y}{(\lambda_xz+\lambda_y)^2}&\text{for }z\ge0\\0&\text{otherwise}\end{cases}

and

f_{Z=XY}(z)=\displaystyle\int_0^\infty\frac1y(\lambda_xe^{-\lambda_xyz})(\lambda_ye^{\lambda_yz})\,\mathrm dy

\implies f_{Z=XY}(z)=\lambda_x\lambda_y\displaystyle\int_0^\infty\frac{e^{-\lambda_x\frac zy-\lambda_yy}}y\,\mathrm dy

I wouldn't worry about evaluating this integral any further unless you know about the Bessel functions.

You might be interested in
Find a formula for the function whose graph is given below.
gayaneshka [121]
Y=(x-2)^2+3
^^^^^^^^^^^^^^^^
5 0
3 years ago
-37 ≥ n - 20 <br> What could n be??
sertanlavr [38]

Answer:

-78

Step-by-step explanation:

-78 - 20 = -98 which is less than -37,

im sorry this answer might be wrong i tried my best

6 0
3 years ago
Read 2 more answers
Three times a number, subtracted from 20, is equal to -7
tia_tia [17]
The answer is 20-3x=-7 explanation
4 0
2 years ago
Number of
jekas [21]

Answer:

The constant rate of change is 3

Step-by-step explanation:

Given

x   y

2   6

4   12

6   18

8   24

Required

Determine the constant rate of change

Represent the constant rate of change with k.

k is calculated using:

k = \frac{y}{x}

When y = 24; x = 8

So, we have:

k = \frac{24}{8}

k = 3

7 0
2 years ago
I really need help with this question
Dominik [7]

Answer:

your screen is blue i can't see the problem dummy

5 0
3 years ago
Other questions:
  • Kramer Fabrications stockholders' equity at the beginning of July 2017 was $600,000. During the month, the company earned net in
    7·1 answer
  • Please help me I'm so confused
    11·1 answer
  • Which operation should you perform first when evaluating the expression 3²+ 2​
    9·2 answers
  • Does a negative divided by a negative equal a positive
    15·2 answers
  • Please help me!<br> Find AC
    14·1 answer
  • HElp pleaseeeeeeeee<br> and please answer properly
    7·1 answer
  • What is the value of the expression 6 3/4 ( -11.5)?
    5·1 answer
  • Need help ASAP! Directions in picture ;)
    15·2 answers
  • The expression (x+3)(x+4), is equivalent to which of the following?
    11·1 answer
  • Which expression is the factorization of x2 10x 21?
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!