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
Rzqust [24]
3 years ago
14

The joint probability density function of X and Y is given by fX,Y (x, y) = ( 6 7 x 2 + xy 2 if 0 < x < 1, 0 < y < 2

0 otherwise. (a) Verify that this is indeed a joint density function. (b) Compute the density function of X. (c) Find P(X > Y ). (d) Find P(Y > 1/2 | X < 1/2). (e) Find E(X). (f) Find E(Y
Mathematics
1 answer:
fredd [130]3 years ago
5 0

I'm going to assume the joint density function is

f_{X,Y}(x,y)=\begin{cases}\frac67(x^2+\frac{xy}2\right)&\text{for }0

a. In order for f_{X,Y} to be a proper probability density function, the integral over its support must be 1.

\displaystyle\int_0^2\int_0^1\frac67\left(x^2+\frac{xy}2\right)\,\mathrm dx\,\mathrm dy=\frac67\int_0^2\left(\frac13+\frac y4\right)\,\mathrm dy=1



b. You get the marginal density f_X by integrating the joint density over all possible values of Y:

f_X(x)=\displaystyle\int_0^2f_{X,Y}(x,y)\,\mathrm dy=\boxed{\begin{cases}\frac67(2x^2+x)&\text{for }0

c. We have

P(X>Y)=\displaystyle\int_0^1\int_0^xf_{X,Y}(x,y)\,\mathrm dy\,\mathrm dx=\int_0^1\frac{15}{14}x^3\,\mathrm dx=\boxed{\frac{15}{56}}

d. We have

\displaystyle P\left(X

and by definition of conditional probability,

P\left(Y>\dfrac12\mid X\frac12\text{ and }X

\displaystyle=\dfrac{28}5\int_{1/2}^2\int_0^{1/2}f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\boxed{\frac{69}{80}}

e. We can find the expectation of X using the marginal distribution found earlier.

E[X]=\displaystyle\int_0^1xf_X(x)\,\mathrm dx=\frac67\int_0^1(2x^2+x)\,\mathrm dx=\boxed{\frac57}

f. This part is cut off, but if you're supposed to find the expectation of Y, there are several ways to do so.

  • Compute the marginal density of Y, then directly compute the expected value.

f_Y(y)=\displaystyle\int_0^1f_{X,Y}(x,y)\,\mathrm dx=\begin{cases}\frac1{14}(4+3y)&\text{for }0

\implies E[Y]=\displaystyle\int_0^2yf_Y(y)\,\mathrm dy=\frac87

  • Compute the conditional density of Y given X=x, then use the law of total expectation.

f_{Y\mid X}(y\mid x)=\dfrac{f_{X,Y}(x,y)}{f_X(x)}=\begin{cases}\frac{2x+y}{4x+2}&\text{for }0

The law of total expectation says

E[Y]=E[E[Y\mid X]]

We have

E[Y\mid X=x]=\displaystyle\int_0^2yf_{Y\mid X}(y\mid x)\,\mathrm dy=\frac{6x+4}{6x+3}=1+\frac1{6x+3}

\implies E[Y\mid X]=1+\dfrac1{6X+3}

This random variable is undefined only when X=-\frac12 which is outside the support of f_X, so we have

E[Y]=E\left[1+\dfrac1{6X+3}\right]=\displaystyle\int_0^1\left(1+\frac1{6x+3}\right)f_X(x)\,\mathrm dx=\frac87

You might be interested in
What is 4 wholes and 49 over 100 in decimal form
SVETLANKA909090 [29]
49/100 is equal to .49 in decimal form, and now we just need to add the 4 wholes.  This is equal to 4.49
5 0
3 years ago
Read 2 more answers
A square pyramid is 6 feet on each side. The height of the pyramid is 4 feet. What is the total area of the pyramid?
Temka [501]
Answer is C, 96 ft squared.
4 0
3 years ago
Read 2 more answers
6 The table below shows the federal income tax rate for wages earned for single adults.
Zielflug [23.3K]

Answer:

F) 35,625

Step-by-step explanation:

Walter is in the 25% range since $47,500 is in-between 36,901 to 89,359

25% = 0.25

47,500 * 0.25 = 11,875

47,500 - 11,875 = $35,625

8 0
3 years ago
Suppose a triangle has two sides of length 33 and 37, and that the angle
marysya [2.9K]

Answer:

c^2=33^2+37^2-2(33)(37)cos120(i don't know for sure tho)

Step-by-step explanation:

LAW OF COSINES

6 0
3 years ago
Need asap for a test !! Which of these tables represents a function?
Dominik [7]
We can’t see the tables unless you upload a picture :)
8 0
3 years ago
Other questions:
  • Geometric objects can be grouped by their _________?
    6·1 answer
  • Is this the correct answer?
    13·1 answer
  • Hello i am bored and wondering if anyone is down to zoom or sum
    14·1 answer
  • using the digits 4,9,7,1 and 5,create a division sentence with two digit divisor and greatest possible quotient
    13·1 answer
  • Helppp PLEASEEEE!!!!!! !!!
    13·1 answer
  • Find the equation of this line.<br> Simplify the fractions<br> (Picture)
    11·1 answer
  • What point is shared by two lines?
    6·2 answers
  • Maths people please write the answers !<br><br><br> Equations- Maths
    6·2 answers
  • Give two ways a debit card can be used.
    5·2 answers
  • Please help i dont know the answer
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!