Mathematical Statistics with Applications
Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
Question
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Chapter 6.4, Problem 29E

a.

To determine

Derive the distribution of W=mV22, the kinetic energy of the molecule.

a.

Expert Solution
Check Mark

Answer to Problem 29E

The probability density function for W=mV22 is fW(w)=1Γ(32)(kT)32w12ewkT,w0.

Explanation of Solution

Calculation:

Gamma distribution:

If the random variable X follows Gamma distribution with (α,β),

  • The probability density function for X is f(x)=1Γ(α)βαxα1exβ,x>0.
  • The mean if X is E(X)=αβ

The probability density function for V is, f(v)=av2ebv2,  v>0.

From the given information, the random variable W is defined as W=mV22.

Consider,

W=mV22mV2=2WV2=2WmV=2Wm

 dVdw=12(2Wm)12×2m=12mw

The probability density function for W is,

fW(w)=f(v)|dvdw|=av2ebv2×|12mw|=a(2wm)2em2kT×(2wm)2×12mw=a2wm32ewkT,w0

The density function is in the form of Gamma density function and the constant ‘a’ is obtained by equating the integral of density function to 1.

Therefore,

01fW(w)dw=101a2wm32ewkTdw=1a2m3201w12ewkTdw=1a2m32×Γ(32)(kT)32=1

                           a2m32=1Γ(32)(kT)32

Therefore, the density function for W is,

 fW(w)=a2wm32ewkT=1Γ(32)(kT)32w12ewkT,w0

b.

To determine

Find the value of E(W).

b.

Expert Solution
Check Mark

Answer to Problem 29E

The value of E(W) is 32kT.

Explanation of Solution

Calculation:

The probability density function for W is fW(w)=1Γ(32)(kT)32w12ewkT,w0.

The probability density function fW(w)=1Γ(32)(kT)32w12ewkT,w0 is similar to the standard Gamma probability density function, f(x)=1Γ(α)βαxα1exβ,x>0 with α=32 and β=kT.

Therefore, the mean of the random variable W is,

E(W)=αβ=32×kT=32kT

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Chapter 6 Solutions

Mathematical Statistics with Applications

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