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

Paint, different types of paints?

different types of paints?

Torsion, assumptions made in the simple torsion theory

assumptions made in the simple torsion theory

Determine concentration of total solids and volatile solids, The following ...

The following test results were obtained for a 100 ml wastewater sample taken at the inlet to wastewater treatment plant.  The tests were performed using the standard methods. Dete

Structural connection loosening - deterioration in steel, Define Structural...

Define Structural connection loosening - Deterioration in Steel? Structural connections joined together by rivets or bolts have a tendency to work loose over an extended period

Strength of materials, distribution of shear stress in triangular section

distribution of shear stress in triangular section

Define project scheduling - types of methods, Define Project Scheduling - T...

Define Project Scheduling - Types of Methods Project Scheduling - Distinction between the Major Types of Schedule Diagrams Precedence Diagrams (PDM)   a) Activ

Stone soling - road pavements, Stone Soling: One of the early practic...

Stone Soling: One of the early practices in road construction was to provide a stone soling 150-225 mm in thickness. The stones are laid by manual labour, and stone spalls ar

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