Week 10 Gaussian Integrals, Boltzmann Factor and Boltzmann Distribution

Question 1

what is the normalised prob dist eqn

Answer 1

ΣPi = 1

Question 2

what is the average of a general function eqn for discrete variables

Answer 2

= ΣPi fi

Question 3

what is the variance of a general function eqn for discrete variables

Answer 3

Var = ΣPi(fi - )^2

Question 4

what is the prob for a gemoetric, poisson and binomial distribution and and conditions

Answer 4

geometric: Pi = (1-λ)λ^i
(00
binomial: Pi = (n choose i)λ^i(1-λ)^(n-i)
0

Question 5

what is the average of a general function eqn for continuous variables

Answer 5

= ∫d^3v ρ(v)f(v)

Question 6

what is the variance of a general function eqn for continuous variables

Answer 6

var = < (f-)^2 > = - ^2

Question 7

what is the general Gaussian integral equation and the initial value

Answer 7

Im = ∫ x^m exp(-αx^2) dx

with I0 = ∫ exp(-αx^2) dx = SQRT(π/a)

Question 8

what is the boltzmann distribution for v

Answer 8

ρ(v) = SQRT(b^3/π^3) exp(-bv^2)