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

Maintenance of airport pavements, Maintenance of Airport Pavements: ...

Maintenance of Airport Pavements: Maintenance Elements The maintenance of airport pavements is an important task of a civil engineer, since the safe movement of aircr

Manhole covers finished of cast iron same as ductile iron, Question Why are...

Question Why are manhole covers finished of cast iron as same are through of ductile iron ? Answer Usually, manholes covers are made of cast iron. Though, in viewpoint of pipe

Define categories of spalls -underwater inspection of bridge, Define catego...

Define categories of Spalls -underwater inspection of bridge? Spalls are classified as follows:  i. Small spalls - up to 25 mm deep or approximately 150 mm in diameter. i

Water waste engineering, wahat is physical characteristic on total suspende...

wahat is physical characteristic on total suspended solid

Kerb overflow weirs – horizontal bars vs. vertical bars, Q. Kerb overflow w...

Q. Kerb overflow weirs – horizontal bars vs. vertical bars? Overflow weirs must be provided for steep roads (longitudinal gradient>5%), flat roads (longitudinal gradient Ess

Dense bituminous macadam, Dense Bituminous Macadam: A Dense Bituminou...

Dense Bituminous Macadam: A Dense Bituminous Macadam (DBM) is a close-graded premix material having a voids content of 5-10 per cent. Two gradings are normally adopted :

Define lateral load - actions in pot bearings, Define Lateral Load - Action...

Define Lateral Load - Actions in POT Bearings? Lateral load is transferred from the piston to the POT by contact between the rim of the piston and the wall of the POT. The cont

Heliports and stol ports, Heliports and Stol Ports: Heliports ...

Heliports and Stol Ports: Heliports Heliports are facilities for landing and take-off of helicopters. They consist of : (a) Landing and take-off area, which contai

Explain dissolved impurities in water, Q. Explain Dissolved Impurities in w...

Q. Explain Dissolved Impurities in water? The impurities which are caused by the action of water and water forming Acids with organic matter and carbon dio

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