Poisson distribution function, Civil Engineering

Assignment Help:

Poisson distribution function:

Let XI, X2, ..., X,& b e n independently and identically distributed random variables each having the same cdf F ( x ). What is the pdf of the largest of the xi'?

Solution:

Let Y = maximum (XI, X2, ..., Xn, )

Since Y ≤ y implies Xl ≤ y, X2 ≤y, ..., Xn ≤ y, we have

Fy(y )= P(Y ≤ y) = P(XI ≤ y,X2 ≤ y, ..., Xn ≤ y )

= P(X1 ≤ y) P(X2 ≤ y) ... P(Xn ≤ y),

since XI, X2, ..., Xn, are independent

= {F(y)}n, since the cdf of each Xi is F(x).

 Hence the pdf of Y is

fy (y) = F'(y) = n{F(y)}n-1 f(y),

where f ( y ) = F'( y ) is the pdf of Xi.

6.2.2 The Method of Probability Density Function (Approach 2)

For a univariate continuous random variable x having the pdf' fx ( x ) and the cdf Fx (x), we have

F'x(x)= (d/dx ) dFx(x) or dFx(x) = fx (x) dx

In other words, differential dFx (x) represents the element of probability that X assumes a value in an infinitesimal interval of width dx in the neighbourhood of X = x.

For a one-to-one transformation y = g ( x ), there exists an inverse transformation x = g - 1 ( y ), so that under the transformation as x changes to y, dx changes to dg-1(y)/dy and

dF (x) = f(x) dx = fx (g-1(y))¦dg-1(y)/dy¦ dy

The absolute value of dg-1(y)/dy is taken because may be negative and fx ( x ) and d Fx ( x ) are always positive. As X, lying in an interval of width dx in the neighbourhood of X = x, changes to y, that lies in the corresponding interval of width dy in the neighbourhood of Y = y, the element of probability dFx ( x ) and dFy ( y ) remain the same where Fy ( y ) is 1 cdf of Y. Hence

dFy(y) = dFx(x) = fx(g-1(y)) ¦ dg-1(y)/dy¦ dy

and

fy(y) = d/dy Fy(y) = fx (g-1 (y)) ¦ dg-1(y)/dy¦                                         (6.2)

Equation (6.2) may be used to find the pdf of a one to one function of a random variable. The method could be generalized to the multivariate case to obtain the result that gives the joint pdf of transformed vector random variable Y under the one to one transformation Y - G ( X ) , in terms of the joint pdf of X The generalized result is stated below

fy(y) = fx(G-1 (y))1/¦J¦

where the usual notations and conventions for the Jacobian J = ¦ð y/ð x¦ are assumed

Remarks:

This technique is applicable hust to continuous random variables and only if the functions of random variable Y = G (X) define a one to one transformation of the region where the pdf of X is non zero.


Related Discussions:- Poisson distribution function

Soils, how to classify soils materials

how to classify soils materials

Widening on curves, Widening on Curves: Pavements have to be widened ...

Widening on Curves: Pavements have to be widened on curves because of the fact that : (a) the vehicle has a wheel base and overhang on the front and rear, and the vehicle

Advantages of unbonded post-tensioning systems, Advantages of unbonded post...

Advantages of unbonded post-tensioning systems Similarly, some advantages of unbonded post-tensioning systems over bonded post-tensioning ones may be written as under :  (a)

State the coulomb earth pressure theory, A masonry wall with vertical back ...

A masonry wall with vertical back has a backfill 5 m behind it. The ground level is horizontal at the top and the ground water table is at ground level. Measured the horizontal pre

Material engineering assignment, select three materials metal and non meala...

select three materials metal and non mealat at least one of the selected materials shold change its property

What is critical steel ratio, Q. What is Critical steel ratio? Critica...

Q. What is Critical steel ratio? Critical steel ratio - only consider 250mm of concrete from outer face The purpose of critical steel ratio is to control cracking pattern

Concrete technology, why strength of transition zone is less than bulk past...

why strength of transition zone is less than bulk paste

Necessity for road drainage - road drainage, Necessity for Road Drainage: ...

Necessity for Road Drainage: Poor drainage can cause damage to the various components of a road in the following manner: (a) Water can enter the various pavement layers th

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd