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
Maslowich
3 years ago
10

If s is an increasing function, and t is a decreasing function, find Cs(X),t(Y ) in terms of CX,Y .

Mathematics
1 answer:
Sedbober [7]3 years ago
7 0

Answer:

C(X,Y)(a,b)=1−C(s(X),t(Y))(a,1−b).

Step-by-step explanation:

Let's introduce the cumulative distribution of (X,Y), X and Y :

F(X,Y)(x,y)=P(X≤x,Y≤y)

  • FX(x)=P(X≤x)
  • FY(y)=P(Y≤y).

Likewise for (s(X),t(Y)), s(X) and t(Y) :

F(s(X),t(Y))(u,v)=P(s(X)≤u

  • t(Y)≤v)
  • Fs(X)(u)=P(s(X)≤u)
  • Ft(Y)(v)=P(t(Y)≤v).

Now, First establish the relationship between F(X,Y) and F(s(X),t(Y)) :

F(X,Y)(x,y)=P(X≤x,Y≤y)=P(s(X)≤s(x),t(Y)≥t(y))

The last step is obtained by applying the functions s and t since s preserves order and t reverses it.

This can be further transformed into

F(X,Y)(x,y)=1−P(s(X)≤s(x),t(Y)≤t(y))=1−F(s(X),t(Y))(s(x),t(y))

Since our random variables are continuous, we assume that the difference between t(Y)≤t(y) and t(Y)<t(y)) is just a set of zero measure.

Now, to transform this into a statement about copulas, note that

C(X,Y)(a,b)=F(X,Y)(F−1X(a), F−1Y(b))

Thus, plugging x=F−1X(a) and y=F−1Y(b) into our previous formula,

we get

F(X,Y)(F−1X(a),F−1Y(b))=1−F(s(X),t(Y))(s(F−1X(a)),t(F−1Y(b)))

The left hand side is the copula C(X,Y), the right hand side still needs some work.

Note that

Fs(X)(s(F−1X(a)))=P(s(X)≤s(F−1X(a)))=P(X≤F−1X(a))=FX(F−1X(a))=a

and likewise

Ft(Y)(s(F−1Y(b)))=P(t(Y)≤t(F−1Y(b)))=P(Y≥F−1Y(b))=1−FY(F−1Y(b))=1−b

Combining all results we obtain for the relationship between the copulas

C(X,Y)(a,b)=1−C(s(X),t(Y))(a,1−b).

You might be interested in
What is the distance between points E and G?
Snowcat [4.5K]
The answer you are looking for is 12.0
8 0
3 years ago
If an airplane’s speed is 850 miles/hour ? What is it’s speed in meters/seconds
Rasek [7]

Answer:

379.984 meters per second.

5 0
3 years ago
Find the gradient of the line segment between the points (3,2) and (2,5).
Alika [10]

The answer is -3, hope this helps!

5 0
3 years ago
Explain now!!! And tell how
Elenna [48]

Answer:

1.33 ounces

Step-by-step explanation:

Divide the food by the cats. 8/6=4/3≈1.33 ounces.

4 0
3 years ago
Read 2 more answers
Plz help<br><br> (2x/0.25)−0.525=7.475<br><br> what's x?
Jlenok [28]

Answer:

i used calculator and got 1

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Other questions:
  • (6•8)+(4•10)+(4•6)+(4•8)​
    13·2 answers
  • Need help fast, being timed!!
    5·1 answer
  • If q is between 0 and 90 and tan(theta)= 7/8, find costheta
    7·1 answer
  • 6th 7th 8th
    15·1 answer
  • The sum of the following algebraic expression 2x + 15, 7-8x and 3x - 41 is 30 find the value of x..​
    11·2 answers
  • Here are some cards with angles written on them.
    14·2 answers
  • On his lunch break, Calvin purchases a burger and a drink. The total order is $8.50, of which $1.08 is tax. How much was Calvin'
    6·2 answers
  • Award brainly for random response! Ü
    7·2 answers
  • Which expression is equivalent to z2 + z + 4z3 + 4z2 ?
    12·1 answer
  • Which statement describes the function y = axn when a = 1 and n is odd?
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!