Let X1;X2;X3;...;X6 be a random sample from a distribution with the following probability desity function:
Fx(x) = (1+4x)/3 for 0<x<1
= 0 for x=<0; x>=1.
(a) Determine the joint p.d.f of Y1 and Y6 where Y1<Y2<...Y6 are the order statistics.
(b) Let R= Y6-Y1 and Z=Y1 in the joint p.d.f in (a).
(i) Find the inverses of the
functions for r and z.
(ii) Compute the Jacobian.
(iii) Determine the joint
probability density
function of R and Z.
(iv) Write down the integral you
would have to solve to
determine the p.d.f of R.
Do not solve the integral.
To determine the joint probability density function (pdf) of the order statistics Y1 and Y6, we need to first find the cumulative distribution function (CDF) and then differentiate it to obtain the joint pdf.
(a) Determine the joint pdf of Y1 and Y6:
1. The probability that Y1 takes on a value less than y is equal to the probability that at least one of the X's in the sample takes on a value less than y. So, P(Y1 < y) can be calculated as follows:
P(Y1 < y) = 1 - P(X1 >= y, X2 >= y, ..., X6 >= y)
2. Since the Xi's are independent, we can calculate the probability as the product of their individual CDFs:
P(Y1 < y) = 1 - (1 - Fx(y))^6
3. Taking the derivative with respect to y, we get the joint pdf of Y1 and Y6:
fY1,Y6(y1, y6) = 6*(1 - Fx(y1))^(5)*(Fx(y6) - Fx(y1))^(1) * fX(y1) * fX(y6)
(b) Now, let's find the joint pdf of R (Y6 - Y1) and Z (Y1):
(i) Finding the inverses of the functions for r and z:
- To find the inverse of R, solve the equation r = y6 - y1 for y6 in terms of r and y1: y6 = r + y1
- To find the inverse of Z, it is already in terms of Y1.
(ii) Compute the Jacobian:
The Jacobian for the transformation (R, Z) -> (Y1, Y6) can be found by taking the determinant of the matrix of partial derivatives:
J = |d(y6)/dr d(y6)/dz|
|d(y1)/dr d(y1)/dz|
Since d(y6)/dz = 0 (no dependence of y6 on z), the Jacobian simplifies to:
J = |1 0|
|1 1| = 1
(iii) Determine the joint pdf of R and Z:
Using the transformation theorem, the joint pdf of R and Z can be found as:
fR,Z(r, z) = fY1,Y6(y1, y6) * |J|
Substituting r + z for y6 in fY1,Y6(y1, y6), we have:
fR,Z(r, z) = 6*(1 - Fx(y1))^(5)*(Fx(r + z) - Fx(y1))^(1) * fX(y1) * 1
(iv) Write down the integral to determine the pdf of R:
To find the pdf of R, we need to integrate the joint pdf of R and Z with respect to Z, while keeping R constant:
fR(r) = ∫[fR,Z(r, z)] dz with limits of integration suitable for the range of Z
The specific integral will depend on the range of values for Z.
Note: Solving the integral and obtaining the final results will require additional calculations depending on the precise form of the distributions and ranges of Z and R.