2.12 Laplace transforms

f(t) = F(s) = L{f(t)} = 0

∞ f(t)e−st dt

(2.524)

δ(t) = 1

(2.525)

1

= 1/s

(s > 0)

(2.526)

tn

=

n!

(s > 0, n > −1)

(2.527)

sn+1

t1/2

=

π

(2.528)

4s3

π

t−1/2

=

(2.529)

s

eat

1

=

(s > a)

(2.530)

s− a

teat

=

1

(s > a)

(2.531)

(s− a)2

(1

−

at)e−at

=

s

(2.532)

+ a)2

(s

t2e−at

=

2

(2.533)

+ a)3

(s

sinat =

a

(s > 0)

(2.534)

s2 + a2

cosat =

s

(s > 0)

(2.535)

s2 + a2

sinhat =

a

(s > a)

(2.536)

s2 − a2

coshat =

s

(s > a)

(2.537)

s2

a2

e−bt sinat =

− a

(2.538)

+ b)2 + a2

(s

e−bt cosat =

s+ b

(2.539)

+ b)2 + a2

(s

e−atf(t) = F(s+ a)

(2.540)

2.13 Probability and statistics

57

Discrete statistics

2

1

N

xi

data series

Mean

·

series length

N

x = N i=1 xi

(2.541)

mean value

−

N

var[

]

unbiased

1

Variancea

var[x] =

N

1

(xi − x )2

(2.542)

·

variance

i=1

Standard

σ[x] = (var[x])1/2

(2.543)

σ

standard

deviation

deviation

−

−

N

3

Skewness

skew[x] =

N

xi − x

(2.544)

(N

1)(N

2)

i=1

σ

1

Kurtosis

kurt[x]

xi − x

3

(2.545)

N i=1

σ

−

N

(xi − x )(yi − y )

x,y

data series to

Correlation

correlate

r =

i=1

(2.546)

N (xi

y )2

coe cientb

−

x )2

N

(yi

−

r

correlation

i=1

i=1

coe cient

distribution

pr(x)

mean

variance

domain

n

binomial

Binomial

xn px(1 − p)n−x

np

np(1 − p)

(x = 0,1,... ,n)

(2.547)

x

coe cient

Geometric

(1

−

p)x−1p

1/p

(1

−

p)/p2

(x = 1,2,3,...)

(2.548)

Poisson

λx exp(−λ)/x!

λ

λ

(x = 0,1,2,...)

(2.549)

distribution

pr(x)

mean

variance

domain

Uniform

1

a+ b

(b− a)2

(a

≤

x

≤

b)

(2.550)

b

−

a

2

12

1/λ2

Exponential

λexp(−λx)

1/λ

(x ≥ 0)

(2.551)

Normal/

1

exp

−(x− µ)2

µ

σ2

(

< x <

)

(2.552)

Gaussian

σ√2π

2σ2

−∞

∞

Chi-squared

a

e−x/2x(r/2)−1

r

2r

(x ≥ 0)

(2.553)

2r/2Γ(r/2)

σ2

2σ2

(

−

4 !

≥

Rayleigh

x

exp

−x2

σ π/2

2σ2

1

π

(x

0)

(2.554)

Cauchy/

a

(none)

(none)

(−∞ < x < ∞)

(2.555)

Lorentzian

π(a2 + x2)

$

pr

probability density

exp

−

1 (x

µ)C−1(x

µ)T

k

number of dimensions

Density function

pr(x) =

2

#

(2π)k/−2[det(C)]1/−2

C

covariance matrix

(2.556)

x

variable (k dimensional)

µ

vector of means

T

transpose

Mean

µ = (µ1,µ2,... ,µk)

(2.557)

det

determinant

µi

mean of ith variable

Covariance

C = σij = xixj

− xi

xj

(2.558)

σij

components of C

Correlation

r =

σij

(2.559)

r

correlation coe cient

coe cient

σiσj

1/2

xi

normally distributed deviates

Box–Muller

x1 = (−2lny1)1/2 cos2πy2

(2.560)

yi

deviates distributed

transformation

x2 = (−2lny1)

sin2πy2

(2.561)

uniformly between 0 and 1