Lecture 3 Flashcards
Mathematical statistics
concerned with theoretical foundations
Applied statistics
concerned with modelling data and the errors, or uncertainties in our observations
statistical error
Uncertainty in the measurement of a physical quantity that is essentially unpredictable
Systematic error
Uncertainty in the measurement of a physical quantity that is always systematically too large or too small. Measurement is biased.
Probability definition
Probability measures our degree of belief that something is true.
probability density function definition
measured continuous variables which can take on infinitely many possible values
how to compute probabilities
Prob(a<x<b) =a ∫b p(x)dx
how to normalise
- ∞ ∫∞ p(x)dx = 1
Poisson PDF
p(r) = Rτ^r-e^(Rτ)/r!
Uniform PDF
p(x) = { 1/(b-a) a<X<b
{ 0 otherwise
Central or Normal PDF
p(x) = 1/√2π σexp[-1/2(x-μ/σ)^2]
Cumulative distribution function CDF
P(a) = - ∞ ∫a p(x)dx = Prob (x<a)
Moment of a discrete case PDF
b Σ x=a x^n p(x) Δx
Moment of a continuous case PDF
a ∫ b x^n p(x) dx
The variance of a discrete case PDF
b Σ a p(x)(x-<x>)^2Δx</x>
The variance of a continuous case PDF
a ∫ b p(x)(x-<x>)^2dx</x>
In general the variance of a PDF is
var[x] = <x^2> - <x>^2</x>
The median of a CDF
divides the CDF in half
=0.5
P(xmed) = 0.5
The mode is
the value of x for which the pdf is a maximum
For a normal PDF in general
mean = median = mode = μ
to find the probability of observing less than x of a Poisson distribution
p(<x) = p(0) + p(1) + p(2) +…+ p(x)
the mean
is the first moment of
(-∞ ∫ ∞) x p(x) dx
how to generate a random sample drawn from a corresponding pdf, using the probability integral transform
Generate a random sample y from U [0,1]
Compute y = P(x)
the xi will represent a sample drawn from p(x) as shown
diagram
proof the mean is = μ
E(N) = Np(N)
= (inf Σ N= 0) Nμ^ Ne^-μ/N!
0 + (inf Σ N= 0) Nμ^N e^-μ/N!
= μ
variance^2
(-∞ ∫ ∞) (x-mean)^2 dx
