Formula

Question 1

expectation value of the number of photons

Answer 1

E(N) = <N> = Rτ</N>

Question 2

Poisson distribution probability of receiving N photons in time τ

Answer 2

p(N) = [(Rτ)^(N) e^(-Rτ)]/N!

where Rτ = µ

Question 3

variance of N

Answer 3

var(N) = E{[N-E(N)]^2}

var(N) = Rτ

Question 4

arrival rate

Answer 4

R(hat) = N(obs)/τ

Question 5

probability density function

Answer 5

Prob(a ≤ x ≤ b) =(b ∫ a) p(x)dx

Question 6

normalisation

Answer 6

(∞ ∫ -∞) p(x) dx = 1

Question 7

uniform pdf

Answer 7

p(x) = { 1/(b-a) a < X < b
{ 0 otherwise

Question 8

central/normal or gaussian pdf

Answer 8

p(x) = 1/√2πσ …

Question 9

CDF

Answer 9

P(x’) = (x’ ∫ -∞) p(x) dx

P(-∞) = 0 P(∞) = 1

Question 10

the nth moment of a pdf discrete case

Answer 10

<x^(n)> = (b Σ x=a) x^(n) p(x)Δx

Question 11

the nth moment of a pdf continuous case

Answer 11

<x^(n)> = (b ∫ a) x^(n) p(x) dx

Question 12

1st moment

Answer 12

= mean or expectation value

Question 13

2nd moment

Answer 13

= mean square

Question 14

variance : discrete case

Answer 14

discrete case 2nd moment - 1st moment^2

Question 15

variance: continuous case

Answer 15

continuous case 2nd moment - 1st moment^2

Question 16

median

Answer 16

P(x(med)) = (x(med) ∫ -∞) p(x’) dx’ = 0.5

Question 17

sample mean

Answer 17

µ(hat) = 1/M (M Σ i =1) x(i)

Question 18

variance of sample mean

Answer 18

var(µ(hat)) = σ^2/M

Question 19

bivariate normal distirbution

Answer 19

p(x,y) on formula sheet

Question 20

quadratic form

Answer 20

Q(x,y) on formula sheet

Question 21

correlation coefficienct

Answer 21

known as p

satistifies E = pσ(x)σ(y)

on formula sheet

Question 22

covariance

Answer 22

cov(x,y) on formula sheet

Question 23

Pearsons product-moment correlation coefficient

Answer 23

r on the formula sheet in the correlation and covariance section

Question 24

ordinary linear least squares

Answer 24

y(i) on formula sheet

Question 25

likelihood function

Answer 25

L = (n Π i=1) p(x(i))

Question 26

chi2

Answer 26

on formula sheet

Question 27

poisson distribution k

Answer 27

= 1

Question 28

normal distirbution k

Answer 28

= 2

Question 29

number of degrees of freedom

Answer 29

r = N - k - 1

Question 30

reduced chi squared

Answer 30

on formula sheet

Question 31

chi2 pdf

Answer 31

p(v)(chi2)

where v is the degrees of freedom

Question 32

P-value

Answer 32

1-P(chi2(obs))

Question 33

variable transformations

Answer 33

p(y)dy = p(x)dx

p(y) = p(x(y))/|dy/dx|

Question 34

power spectral density

Answer 34

= total power = (∞ ∫ -∞) |h(t)|^2 dt = (∞ ∫ -∞) |H(f)|^2 df

parsevals theorem

Question 35

critical frequency

Answer 35

H(f) = 0 for all |f| ≥ f(c)

Nquist-Shannon Sampling Theorem

Question 36

alias effect

Answer 36

if f(s) < 2f

Question 37

Nquist-Shannon Sampling Theorem

Answer 37

T < 1/2f(c)

or f > 2f(c)

Question 38

S = χ^2(a,b)

Answer 38

S = (N Σ i=1) ε^2(i)

where y(i) = a + bx(i) + ε(i)

so ε(i)^2 = [y(i) - a - bx(i)]^2

Question 39

mean =

Answer 39

total counts / area

Question 40

p(<5) =

Answer 40

p(0) + p(1) + p(2) + p(3) + p(4)

Question 41

maximum likelihood

Answer 41

l = ln(L)

differentiate and set equal to zero

Question 42

y ~ N[0,1]

Answer 42

normal pdf with mean zero and standard deviation unity

Question 43

if we want z ~ N[µ, σ]

Answer 43

z = µ+σy

and p(y) =

Question 44

x ~ U[0,1]

and p(x) = { 1 for 0<x<1
{ otherwise

Answer 44

y = a+(b-a)x

Question 45

P(exp)(x) =

Answer 45

(x ∫ 0) p(exp) dx

Question 46

variance =

Answer 46

(standard deviation)^2
σ^2

σ = √variance

Question 47

fourier transform of d(t) = δ(t-1) + δ(t+1)

Answer 47

exp[2πif] + exp[-2πif]

Question 48

standard deviation =

in poisson case

Answer 48

õ

Question 49

RMS =

Answer 49

√{A^2[f(upp)-f(lower)]}

where A a constant amplitude = white noise