Basic Statistical Toolbox Flashcards

1
Q

mathematical statistics

A

concerned with theoretical foundations (probability theory)

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2
Q

applied statistics

A

concerned with modelling of data and the errors in our observations to make inferences about the physical system we are observing

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3
Q

statistical error

A

Uncertainty in the measurement of a physical quantity that is essentially unpredictable – just
as likely to yield a measurement that is too large as one that is too small.

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4
Q

‘common sense principle’

A

if we repeat our measurements many times and average the results then average length = ‘true’ length

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5
Q

systematic error

A

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

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6
Q

Note that here the systematic error enters not when we make our measurements, but

A

when we analyse them

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7
Q

Systematic flaws in our data analysis methods, rather than in
our data themselves, are just as serious but

A

easier to fix

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8
Q

plausible reasoning

A

probability measures our degree of belief that something is true

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9
Q

prob(X)=1

A

certain that X is true

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10
Q

prob(X)=0

A

certain that X is false

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11
Q

In astronomy we generally measure continuous variables which can
take on

A

infinitely many possible values

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12
Q

with infinitely many possible values, p(X) is no longer a probability but a

A

probability density function

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13
Q

probabilities are never

A

negative
p(x)>or=0 for all x

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14
Q

we compute probabilities by

A

measuring the area under the pdf curve

ie integral between a and b of p(x) dx

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15
Q

normalisation

A

integral between -infinity and infinity =1

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16
Q

important pdfs:

A
  1. poisson
  2. uniform
  3. central/normal/gaussian
17
Q

examples of poisson pdfs

A

number of photons counted by CCD
number of galaxies counted by galaxy survey
photons in laser beam

18
Q

poisson pdf assumes

A

detections are independent and there is a constant rate μ

19
Q

uniform pdf

A

p(x)=1/b-a when a<x<b
o otherwise

20
Q

what is a cdf

A

cumulative distribution function

21
Q

cdf P(a)=

A

integral from -infinity to a of p(x)dx = prob(x<a)

22
Q

the nth moment of a pdf is defined as (discrete case)

A

sum from x=a to x= of x^np(x)delta x

23
Q

nth moment of a pdf for the continuous case

A

integral from a to b of x^np(x)dx

24
Q

1st moment is called the

A

mean or expectation value

25
Q

second moment is called

A

the mean square

26
Q

the variance for the discrete case is defined as

A

sum between a and b of p(x)(x-<x>)^2delta x</x>

27
Q

the variance for the continuous case

A

integral between a and b of p(x)(x-<x>)^2dx</x>

28
Q

the variance is often written as

A

σ^2

29
Q

σ=sqrt(σ^2) is called

A

the standard deviation

30
Q

in general, var[x]=

A

<x^2>-<x>^2</x>

31
Q

the median divides the CDF into

A

two equal halves

32
Q

prob(x<xmed)=

A

prob(x>xmed)=0.5

33
Q

the mode is the values of x for which

A

the pdf is a maximum

34
Q

for a normal pdf, mean=

A

median=mode=μ

35
Q
A