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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 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 N(A) P N(tot ) 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-------- … -2 -1 0 +1 +2 +3 Consider the probability of ending up at a particular position after n steps: 0 Position … +3 +2 +1 0 -1 -2 -3 1 2 3 … Number of steps 1/8 3 heads 3/8 2 heads 3/8 1 head 1/8 0 heads ¼ ☺ ½ 2/4 ½ ¼ P(m,n) = Probability of ending up at position m after n steps = In n steps, # paths leadin position to n g # possible paths leadin an to position y g # possible paths = 2 n (each step has 2 outcomes) # paths leading to a particular position m = # of ways of getting k heads n n ! k k !(nk)! Where Binomial coefficient, or “n over k” n! = n(n-1)(n-2) … (2)(1) recall 0! =1 1! = 1 “n-factorial” NOTE: The binomial coefficients n k appear in the expansion a b n n k 0 n k k a b n k Does this formula work in our 1-dimensional random walk example? Let n=3 n 2 =8 suppose k =2 after 3 steps, there are 8 possible paths (Two heads): 3 3 21 3 2! 2 so P 3/8 More generally, for each individual event, OK! P 1/2 Binomial Distribution Probability of getting k successes out of n tries, when the probability for success in each try is p n-k n k P(k,n, p) p) k p (1 n n! k k!(nk)! MEAN: If we perform an experiment n times, and ask how many successes are observed, the average number will approach the mean, k P ( k , n , p ) np n k 0 VARIANCE: (k - ) P(k,n,p) 2 n 2 k 0 n p 1 -p) ( 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 n-k n k P(k,n, p) p) k p (1 n n! k k!(nk)! 3 SO… 10 3 1 5 1 10 ! P ( 3 , 1 ,) 0 0 .1 5 63 (1 ! 30 )!6 6 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 P (k , ) k e k! 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 ∞ 2 kμ 1 1 P ( k , , ) exp 2 2 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