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

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

One-Dimensional Random Walk Flip a coin -- move +1 steps if heads, -1 steps if tails …---x x x x x x … Consider the probability of ending up at a particular position after n steps: …  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

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

NOTE : The binomial coefficients appear in the expansion

Does this formula work in our 1-dimensional random walk example? Let n = 3 2 n = 8 after 3 steps, there are 8 possible paths suppose k =2 (Two heads): OK! More generally, for each individual event,

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,

VARIANCE:

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…

The binomial distribution for n = 10, p=1/6. The mean value is 1.67 The standard deviation (sqrt of variance) is 1.18

Poisson Distribution Assymptotic limit of the binomial distribution for p << 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 k>10 For example, P(20,1.67) = 2.2x10 -15

Gaussian Distribution Gives the most probable estimate of the true mean, µ, of a random sample of observations, as n  ∞ The normal, or Gaussian, distribution In units of standard deviation, σ With origin at the mean value µ The area under the curve = 1

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)

 Animated Gif  Binomial distribution Applet