Lecture 3

Question 1

Mathematical statistics

Answer 1

concerned with theoretical foundations

Question 2

Applied statistics

Answer 2

concerned with modelling data and the errors, or uncertainties in our observations

Question 3

statistical error

Answer 3

Uncertainty in the measurement of a physical quantity that is essentially unpredictable

Question 4

Systematic error

Answer 4

Uncertainty in the measurement of a physical quantity that is always systematically too large or too small. Measurement is biased.

Question 5

Probability definition

Answer 5

Probability measures our degree of belief that something is true.

Question 6

probability density function definition

Answer 6

measured continuous variables which can take on infinitely many possible values

Question 7

how to compute probabilities

Answer 7

Prob(a<x<b) =a ∫b p(x)dx

Question 8

how to normalise

Answer 8

• ∞ ∫∞ p(x)dx = 1

Question 9

Poisson PDF

Answer 9

p(r) = Rτ^r-e^(Rτ)/r!

Question 10

Uniform PDF

Answer 10

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

Question 11

Central or Normal PDF

Answer 11

p(x) = 1/√2π σexp[-1/2(x-μ/σ)^2]

Question 12

Cumulative distribution function CDF

Answer 12

P(a) = - ∞ ∫a p(x)dx = Prob (x<a)

Question 13

Moment of a discrete case PDF

Answer 13

b Σ x=a x^n p(x) Δx

Question 14

Moment of a continuous case PDF

Answer 14

a ∫ b x^n p(x) dx

Question 15

The variance of a discrete case PDF

Answer 15

b Σ a p(x)(x-<x>)^2Δx</x>

Question 16

The variance of a continuous case PDF

Answer 16

a ∫ b p(x)(x-<x>)^2dx</x>

Question 17

In general the variance of a PDF is

Answer 17

var[x] = <x^2> - <x>^2</x>

Question 18

The median of a CDF

Answer 18

divides the CDF in half

=0.5

P(xmed) = 0.5

Question 19

The mode is

Answer 19

the value of x for which the pdf is a maximum

Question 20

For a normal PDF in general

Answer 20

mean = median = mode = μ

Question 21

to find the probability of observing less than x of a Poisson distribution

Answer 21

p(<x) = p(0) + p(1) + p(2) +…+ p(x)

Question 22

the mean

Answer 22

is the first moment of

(-∞ ∫ ∞) x p(x) dx

Question 23

how to generate a random sample drawn from a corresponding pdf, using the probability integral transform

Answer 23

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

Question 24

proof the mean is = μ

Answer 24

E(N) = Np(N)

= (inf Σ N= 0) Nμ^ Ne^-μ/N!

0 + (inf Σ N= 0) Nμ^N e^-μ/N!

= μ

Question 25

variance^2

Answer 25

(-∞ ∫ ∞) (x-mean)^2 dx