Lecture 1: Random Walks, Distribution Functions

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Lecture 1 Random Walks, Distribution Functions

• Probability and Statistics Fundamental in most
  parts of astronomy
• Examples
• Description of systems
• motion of molecules in an ideal gas
• motion of stars in a globular cluster
• description of radiation field
• Stochastic phenomena
• Radiative Transfer
• Estimating experimental uncertainties

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Probability
The probability P of a particular outcome of an
experiment is an estimate of the likely FRACTION
of a number of repeated observations which lead
to a particular outcome
Where N(A) of outcomes A
N(tot) Total of possible outcomes
Example Flip a coin, A heads, then P 1/2
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One-Dimensional Random Walk
Flip a coin -- move 1 steps if heads, -1 steps
if tails
---x -------- x -------- x -------- x --------
x -------- x-------- -2 -1
0 1 2 3
Consider the probability of ending up at a
particular position after n steps
0 1 2 3 ?Number of
steps 3 1/8
?3 heads 2 ¼ 1
½ 3/8 ?2 heads 0 ?
2/4 -1 ½
3/8 ?1 head -2 ¼
-3 1/8
?0 heads
Position
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P(m,n) Probability of ending up at position m
after n steps
In n steps, possible paths 2 n
(each step has 2 outcomes)
paths leading to a particular position m

of ways of getting k heads
Binomial coefficient, or n over k
Where n! n(n-1)(n-2) (2)(1)
n-factorial
recall 0! 1
1! 1
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NOTE The binomial coefficients
appear in the expansion
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Does this formula work in our 1-dimensional
random walk example? Let n 3 2n
8 after 3 steps, there are
8 possible paths suppose k 2 (Two heads)
OK!
More generally, for each individual event,
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Binomial Distribution
Probability of getting k successes out of n
tries, when the probability for success in each
try is p
MEAN If we perform an experiment n times, and
ask how many successes are observed, the
average number will approach the mean,
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VARIANCE
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Example Suppose we roll a die 10 times.
What is the probability that we roll a 2
exactly 3 times? If we throw the die once, the
probability of getting a 2 is p 1/6 For n
10 rolls of the die, we expect to get k 3
successes with probability
SO
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The binomial distribution for n 10, p1/6. The
mean value is 1.67 The standard deviation (sqrt
of variance) is 1.18
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Poisson Distribution

• Assymptotic limit of the binomial distribution
  for p ltlt 1
• Large n, constant mean
• ? small samples of
  large populations

The poisson distribution P(k,1.67). The mean is
1.67, standard deviation is 1.29 Similar to
binomial distribution, but is defined for
kgt10 For example, P(20,1.67) 2.2x10-15
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Gaussian Distribution
Gives the most probable estimate of the true
mean, µ, of a random sample of observations, as
n ? 8
The normal, or Gaussian, distribution In units of
standard deviation, s With origin at the mean
value µ The area under the curve 1
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Random Walk
How far do you get from the origin after n
steps? Plot distance from origin as a
function of n number of steps As n
increases, you stray further and further from the
starting point during an individual
experiment On the other hand, the average
distance of all experiments is ZERO ? as many
experiments end up ve as ve
Root-mean-square distance After n steps,
each of unit distance RMS distance
traveled sqrt(n)
(can
show)
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• Animated Gif
• Binomial distribution Applet