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]
2 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]2 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
What is the answer for 3x=1.2+1.2+1.2
Savatey [412]
The answer is 1.2 b/c you have to solve x by simplifying both sides of the equation and isolating the variable.
5 0
3 years ago
GEOMETRY PROOFS:
Marina CMI [18]

Answer:

Statement 2: ∠BAC = ∠CZY

Reason 3: Vertically opposite angles

Statement 4: YC = CB

Reason 5: AAS (Angle, Angle, Side)

Reason 6: Corresponding sides, congruent triangles

8 0
3 years ago
A survey of 46 college athletes found that 24 played volleyball, while 22 played basketball. a) If we pick one athlete survey pa
PSYCHO15rus [73]

Answer:

A) \dfrac{11}{23}

B) \dfrac{88}{345}

Step-by-step explanation:

A survey of 46 college athletes found that

  • 24 played volleyball,
  • 22 played basketball.

A) If we pick one athlete survey participant at random,  the probability they play basketball is

P_1=\dfrac{22}{46}=\dfrac{11}{23}

B) If we pick 2 athletes at random (without replacement),

  • the probability we get one volleyball player is \dfrac{24}{46}=\dfrac{12}{23};
  • the probability we get another basketball player is \dfrac{22}{45} (only 45 athletes left).

Thus, the probability we get one volleyball player and one basketball player is

P_2=\dfrac{12}{23}\cdot \dfrac{22}{45}=\dfrac{88}{345}

5 0
3 years ago
F(x)=2x^2+3 and g(x)=x^2-7 find (f+g)(x)
defon

Answer:

3x^2 -4

Step-by-step explanation:

F(x)=2x^2+3 and g(x)=x^2-7

(f+g)(x) =

We add the two functions together

(f+g)(x) =2x^2+3+x^2-7

I like to line them up vertically

(f+g)(x) =

2x^2 +3

+x^2  -7

------------------

3x^2 -4

3 0
2 years ago
Which function has the smallest y-intercept value?
Lerok [7]

Answer:

Could you... put it normally?

Step-by-step explanation:

It's kinda confusing to read, so much understand

8 0
3 years ago
Other questions:
  • Which is the best description of 4 (21x÷7)
    6·1 answer
  • What is the domain of the function f(x)=x−4÷√​6−x​​​ 
    11·2 answers
  • Translate the word phrase into a math expression. 11 times the sum of a number and 15
    13·2 answers
  • Basic math plz help<br> volume of a cylinder with the given height and radius or diameter problems.
    12·1 answer
  • Which number is the smallest? 4.4 ⋅ 10−5, 3.99 ⋅ 10−6, 2.6 ⋅ 102, 1.85 ⋅ 103
    6·2 answers
  • The above figure is a special trapezoid. Explain the relationship between the m∠B and m∠C.
    12·1 answer
  • Lowest common factor of 72 and 108​
    12·1 answer
  • The quotient of a number and 2 is the same as the difference of the number doubled and 3. What is the number?
    8·1 answer
  • Find the value of the variable of Y
    11·2 answers
  • Urgent help……………………:::::::::::
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!