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
What is the equation of the circle with center (5,-4) and radius 4
Ede4ka [16]
16 is the right one
4 0
3 years ago
Please help!! Which of the following is equal to the rational expression when x ≠ 2 or -4? 5(x-2)/(x-2)(x+4)
Stels [109]

Answer:

5 / (x+4)      x ≠2 x≠-4

Step-by-step explanation:

5(x-2)/(x-2)(x+4)

The denominator cannot be zero so x ≠2 x≠-4

Cancel like terms in the numerator and denominator

5 / (x+4)      x ≠2 x≠-4

6 0
3 years ago
1.Write an equation in slope- intercept form of the line that passes through the given point and is parallel to the graph of the
Alja [10]
I dont know all of them sorry. But i think
Its d. If its a list of questions lol. But I could be wrong but i did the math 2 times.
8 0
3 years ago
11y + 19x - 5y +3x= pleas e show working please
garri49 [273]

Answer:

11y + 5y = 16y and 19x + 3x= 22x so 16 y +22x is your answer

Step-by-step explanation:brainliest plzzzz

3 0
2 years ago
Read 2 more answers
5x+21+178/(x-7)<br> How do I solve for domain and range on this equation?
Luda [366]
The domain is easiest. You simply need to find the fractions, and exclude any X that makes the denominator zero. So in this case, your domain is everything except X=7.

For range, you are trying to find out what are the possible answers. Usually the best way to do this is to try x=infinity, and x=negative infinity, which here gives you answers of infinity and negative infinity. Since the function is almost continuous, I think your range is any real number.
3 0
3 years ago
Other questions:
  • Ayúdeme es para mañana
    13·1 answer
  • The U.S. Food and Drug Administration (FDA) performed a study that compared the carbohydrate content (as a percentage of weight)
    5·1 answer
  • XY bisects DF at point E. If DE=2y and EF=8y-3, find the value of DF​
    15·1 answer
  • Last week Jean ran 2 fewer than 4m miles. This week she ran 0.5 miles more than last week. Write and simplify an expression for
    6·1 answer
  • Which of the following equations represents the line with a slope of 2/3 and a y-intercept of 9?
    6·1 answer
  • 19) Suppose that there are 55 Democrats and 45 Republicans in the US Senate. A committee of seven senators is to be formed by se
    11·1 answer
  • Matthew is a single taxpayer who earns $75,000 per year in taxable income working as an accountant. He has $2,000 in long-term c
    12·1 answer
  • Draw the number line to represent.
    15·1 answer
  • Find the value of × in each of the following<br><br> a)5+1=11<br> b) 2x-3=2
    11·1 answer
  • What is the mean, median and mode of the following data?
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!