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
finlep [7]
3 years ago
10

Suppose X and Y are random variables with joint density function. f(x, y) = 0.1e−(0.5x + 0.2y) if x ≥ 0, y ≥ 0 0 otherwise (a) I

s f a joint density function? Yes No (b) Find P(Y ≥ 8). (Round your answer to four decimal places.) Find P(X ≤ 5, Y ≤ 8). (Round your answer to four decimal places.) (c) Find the expected value of X. Find the expected value of Y.
Mathematics
1 answer:
Hatshy [7]3 years ago
5 0

a. f_{X,Y} is a joint density function if its integral over the given support is 1:

\displaystyle\int_{-\infty}^\infty\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\frac1{10}\int_0^\infty\int_0^\infty e^{-x/2-y/5}\,\mathrm dx\,\mathrm dy

=\displaystyle\frac1{10}\left(\int_0^\infty e^{-x/2}\,\mathrm dx\right)\left(\int_0^\infty e^{-y/5}\,\mathrm dy\right)=\frac1{10}\cdot2\cdot5=1

so the answer is yes.

b. We should first find the density of the marginal distribution, f_Y(y):

f_Y(y)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dx=\frac1{10}\int_0^\infty e^{-x/2-y/5}\,\mathrm dy

f_Y(y)=\begin{cases}\dfrac15e^{-y/5}&\text{for }y\ge0\\\\0&\text{otherwise}\end{cases}

Then

P(Y\ge8)=\displaystyle\int_8^\infty f_Y(y)\,\mathrm dy=e^{-8/5}

or about 0.2019.

For the other probability, we can use the joint PDF directly:

P(X\le5,Y\le8)=\displaystyle\int_0^5\int_0^8f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=1+e^{-41/10}-e^{-5/2}-e^{-8/5}

which is about 0.7326.

c. We already know the PDF for Y, so we just integrate:

E[Y]=\displaystyle\int_{-\infty}^\infty y\,f_Y(y)\,\mathrm dy=\frac15\int_0^\infty ye^{-y/5}\,\mathrm dy=\boxed5

You might be interested in
PLZZ HELP
9966 [12]

An arithmetic sequence would have a common difference between successive terms, not the case here.

A geometric sequence has a common ratio; let's check:

\dfrac{1/2}{1/6} = 3

\dfrac{3/2}{1/2} = 3

\dfrac{9/2}{3/2} = 3

That's a common ratio of 3 so as far as we can tell a geometric sequence.

Answer: geometric

3 0
3 years ago
Read 2 more answers
I need help with this please...
Luda [366]
Answer:
I think it should be 56
5 0
2 years ago
Read 2 more answers
Please help for 15 Points
rusak2 [61]

1.) 1:800

2.) 1:40

3.) 2:9

Hope this helps!

8 0
3 years ago
Read 2 more answers
F(x)=4x+2 find f(8)
JulsSmile [24]

Answer:

34

Step-by-step explanation:

Substitute 8 where there is an x

f (8)= 4(8) +2

= 32 +2

=34

8 0
3 years ago
Read 2 more answers
Multiple all the coordinates by 1/5.
Andrew [12]

Answer:

(-1,2) , (2,-1), (2,-1) , (-2,1)

Step-by-step explanation:

-5 x 0.2=-1

5 x 0.2=1

10 x 0.2= 2

-10x 0.2=-2

6 0
3 years ago
Other questions:
  • What is the product of 5 + 2i and 7 + 3i?
    5·2 answers
  • 40 is what percent of 50? 8% 12.5% 80% 125%
    8·2 answers
  • How do you write the expression without a fraction bar; <br> a to the seventh/a to the tenth
    5·1 answer
  • Write the equation of the line given the following. Write the equation in slope-intercept form.
    15·1 answer
  • Which is more 10 pancakes with a diameter of 6 inches or 25 pancakes with a radius of 2 inches or 50 pancakes with a circumferen
    13·1 answer
  • 3. Valentina and Chloe were standing next to each other. Chloe walked south for 86 meters
    11·1 answer
  • HELP I NEED HELP ASAP HELP I NEED HELP ASAP HELP I NEED HELP ASAP HELP I NEED HELP ASAP
    11·1 answer
  • PLS HELP ME I really need help right now (15 points)
    7·2 answers
  • Need help with the problem in the picture thank you :)
    9·1 answer
  • 7,14,15,9,11,14,11,10,17 what is the mode of the list of numbers?
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!