Stochastic Processes for Physicists

Transcription

1 Stochastic Processes for Physicists Understanding Noisy Systems Chapter 1: A review of probability theory Paul Kirk Division of Molecular Biosciences, Imperial College London 19/03/2013

2 1.1 Random variables and mutually exclusive events Random variables Suppose we do not know the precise value of a variable, but may have an idea of the relative likelihood that it will have one of a number of possible values. Let us call the unknown quantity X. This quantity is referred to as a random variable. Probability 6-sided die. Let X be the value we get when we roll the die. Describe the likelihood that X will have each of the values 1,..., 6 by a number between 0 and 1: the probability. If Prob(X = 3) = 1, then we will always get a 3. If Prob(X = 3) = 2/3, then we expect to get a 3 about two-thirds of the time. Paul Kirk 1 of 22

3 1.1 Random variables and mutually exclusive events Mutually exclusive events The various values of X are an example of mutually exclusive events. X can take precisely one of the values between 1 and 6. Mutually exclusive probabilities sum Prob(X = 3 or X A= review 4) = Prob(X of probability = 3) + Prob(X theory = 4). Paul Kirk 2 of 22

4 1.1 Random variables and mutually exclusive events Note: in mathematics texts it is customary to denote the unknown quantity using a capital letter, say X, and a variable that specifies one of the possible values that X may have as the equivalent lower-case letter, x. We will use this convention in this chapter, but in the following chapters we will use a lower-case letter for both the unknown quantity and the values it can take, since it causes no confusion. So, rather than writing Prob(X = 3) or Prob(X = x), we will (in later chapters) tend to write Prob(3) or Prob(x). Paul Kirk 3 of 22

5 1.1 Random variables and mutually exclusive events Note: in mathematics texts it is customary to denote the unknown quantity using a capital letter, say X, and a variable that specifies one of the possible values that X may have as the equivalent lower-case letter, x. We will use this convention in this chapter, but in the following chapters we will use a lower-case letter for both the unknown quantity and the values it can take, since it causes no confusion. So, rather than writing Prob(X = 3) or Prob(X = x), we will (in later chapters) tend to write Prob(3) or Prob(x). Warning: may cause confusion. Paul Kirk 3 of 22

6 1.1 Random variables and mutually exclusive events Continuous random variables For continuous random variables, the probability for X to be within a range is found by integrating the probability density function. Prob(a < X < b) = b a P(x)dx 2 A review of probability theory Figure 1.1. An illustation of summing the probabilities of mutually exclusive events, both for discrete and continuous random variables. Paul Kirk 4 of 22

7 1.1 Random variables and mutually exclusive events Expectation The expectation of an arbitrary function, f (X ), with respect to the probability density function P(X ) is f (X ) P(X ) = The mean or expected value of X is X. P(x)f (x)dx. Paul Kirk 5 of 22

8 1.1 Random variables and mutually exclusive events Variance The variance of X is the expectation of the squared difference from the mean V [X ] = = P(x)(x X ) 2 dx P(x)(x 2 + X 2 2x X )dx = P(x)x 2 dx + = X 2 + ( X 2 = X 2 + X 2 2 X 2 = X 2 X 2 P(x)dx P(x) X 2 dx ) ( 2 X P(x)2x X dx ) P(x)xdx Paul Kirk 6 of 22

9 1.2 Independence Independence Independent probabilities multiply For independent variables, P X,Y (x, y) = P X (x)p Y (y) For independent variables, XY = X Y Paul Kirk 7 of 22

10 1.3 Dependent random variables Dependence If X and Y are dependent, then P X,Y (x, y) does not factor as the product of P X (x) and P Y (y) If we know P X,Y (x, y) and want to know P X (x), then it is obtained by integrating out (or marginalising ) the other variable: P X (x) = P X,Y (x, y)dy Paul Kirk 8 of 22

11 1.3 Dependent random variables Conditional probability densities The probability density for X given that we know that Y = y is written P(X = x Y = y) or P(x y) and is referred to as the conditional probability density for X given Y. P X Y (X = x Y = y) = P X,Y (X = x, Y = y). P Y (Y = y) Explanation: To see how to calculate this conditional probability, we note first that P(x, y) with y = a gives the relative probability for different values of x given that Y = a. To obtain the conditional probability density for X given that Y = a, all we have to do is divide P(x, a) by its integral over all values of x. Paul Kirk 9 of 22

12 1.4 Correlations and correlation coefficients The covariance of X and Y is: cov(x, Y ) = (X X )(Y Y ) = XY X Y. Idea: 1. How can we define what it means for a value x to be bigger than usual? Well, we can see if x > X i.e. if x X > Similarly, we can say that a value x is smaller than usual if x < X i.e. if x X < If x tends to be bigger than usual when y is bigger than usual, then (X X )(Y Y ) will be > 0 The correlation is just a normalised version of the covariance, which takes values in the range 1 to 1: C XY = XY X Y V [X ]V [Y ] Paul Kirk 10 of 22

13 1.5 Adding random variables together When we have two continuous random variables, X and Y, with probability densities P X and P Y, it is often useful to be able to calculate the probability density of the random variable whose value is the sum of them: Z = X + Y. It turns out that the probability density for Z is given by P Z (z) = P X (s z)p Y (s)ds = P X P Y Paul Kirk 11 of 22

14 1.5 Adding random variables together When we have two continuous random variables, X and Y, with probability densities P X and P Y, it is often useful to be able to calculate the probability density of the random variable whose value is the sum of them: Z = X + Y. It turns out that the probability density for Z is given by P Z (z) = P X (s z)p Y (s)ds = P X P Y Paul Kirk 11 of 22

15 1.5 Adding random variables together When we have two continuous random variables, X and Y, with probability densities P X and P Y, it is often useful to be able to calculate the probability density of the random variable whose value is the sum of them: Z = X + Y. It turns out that the probability density for Z is given by P Z (z) = P Z (z) = P X (s z)p Y (s)ds = P X P Y P X,Y (z s, s)ds = P X,Y (s, z s)ds Paul Kirk 11 of 22

16 1.5 Adding random variables together When we have two continuous random variables, X and Y, with probability densities P X and P Y, it is often useful to be able to calculate the probability density of the random variable whose value is the sum of them: Z = X + Y. It turns out that the probability density for Z is given by P Z (z) = P Z (z) = P X (s z)p Y (s)ds = P X P Y P X,Y (z s, s)ds = If X and Y are independent, this becomes: P Z (z) = P X,Y (s, z s)ds P X (z s)p Y (s)ds = P X P Y Paul Kirk 11 of 22

17 1.5 Adding random variables together If X 1 and X 2 are random variables and X = X 1 + X 2, then X = X 1 + X 2, and if X 1 and X 2 are independent, then V [X ] = V [X 1 ] + V [X 2 ]. Mysterious (?) assertion Averaging the results of a number of independent measurements produces a more accurate result. This is because the variances of the different measurements add together. does this make sense? Paul Kirk 12 of 22

18 1.5 Adding random variables together Explanation Assume all measurements have expectation, µ, and variance, σ 2. By the independence assumption, the variance of the average is: Moreover, V V [ N n=1 [ ] [ ] ( Xn X 2 = E n N N 2 E X n N ] = N V n=1 [ ] Xn. N [ ]) 2 Xn = E [ X 2 ] n (E [Xn ]) 2 N N 2 = V [X n] N 2. So, V [ N n=1 X n N ] = N n=1 V [X n ] N 2 = 1 N 2 N n=1 σ 2 = σ2 N. Paul Kirk 13 of 22

19 1.6 Transformations of a random variable Key assertion: If Y = g(x ), then: f (Y ) = x=b x=a P X (x)f (g(x))dx = y=g(b) y=g(a) P Y (y)f (y)dy. Paul Kirk 14 of 22

20 1.6 Transformations of a random variable Key assertion: If Y = g(x ), then: f (Y ) = x=b x=a P X (x)f (g(x))dx = y=g(b) y=g(a) P Y (y)f (y)dy. Given this assumption, everything else falls out automatically: f (Y ) = = x=b x=a y=g(b) y=g(a) P X (x)f (g(x))dx = P X (g 1 (y)) g (g 1 (y)) f (y)dy. y=g(b) y=g(a) P X (g 1 (y))f (y) dx dy dy Paul Kirk 14 of 22

21 1.6 Transformations of a random variable Key assertion: If Y = g(x ), then: f (Y ) = x=b x=a P X (x)f (g(x))dx = y=g(b) y=g(a) P Y (y)f (y)dy. Given this assumption, everything else falls out automatically: f (Y ) = = x=b x=a y=g(b) y=g(a) P X (x)f (g(x))dx = P X (g 1 (y)) g (g 1 (y)) f (y)dy. General result (for invertible g): y=g(b) y=g(a) P Y (y) = P X (g 1 (y)) g (g 1 (y)). P X (g 1 (y))f (y) dx dy dy Paul Kirk 14 of 22

22 1.7 The distribution function The probability distribution function, which we will call D(x), of a random variable X is defined as the probability that X is less than or equal to x. Thus D(x) = Prob(X x) = x P(z)dz In addition, the fundamental theorem of calculus tells us that P(x) = d dx D(x). Paul Kirk 15 of 22

23 1.8 The characteristic function The characteristic function is defined as the Fourier transform of the probability density. χ(s) = The inverse transform gives: P(x) = 1 2π P(x) exp (isx )dx. χ(s) exp ( isx)ds. The Fourier transform of the convolution of two functions, P(x) and Q(x), is the product of their Fourier transforms, χ P (s) and χ Q (s). For discrete random variables, the characteristic function is a sum. In general (for both discrete and continuous r.v. s), we have: χ P (s) = exp (isx) P(X ). Paul Kirk 16 of 22

24 1.9 Moments and cumulants Moment generating function (departure from the book) The moment generating function is defined as: M(t) = exp(tx ), where X is a random variable, and the expectation is with respect to some density P(X ), so that M(t) = = exp(tx)p(x)dx ( 1 + tx + 1 2! t2 x ) P(x)dx = 1 + tm ! t2 m r! tr m r +... where m r = X r is the r-th (raw) moment. Paul Kirk 17 of 22

25 1.9 Moments and cumulants Moment generating function (continued) M(t) = 1 + tm ! t2 m r! tr m r +... It follows from the above expansion that: M(0) = 1 M (0) = m 1 M (0) = m 2. M (r) (0) = m r Paul Kirk 18 of 22

26 1.9 Moments and cumulants Cumulant generating function (continued departure from book) The log of the moment generating function is called the cumulant generating function, R(t) = ln(m(t)). By the chain rule of differentiation, we can write down the derivatives of R(t) in terms of the derivatives of M(t), e.g. R (t) = M (t) M(t) R (t) = M(t)M (t) (M (t)) 2 (M(t)) 2 Note that R(0) = 1, R (0) = M (0) = m 1 = µ, R (0) = M (0) (M (0)) 2 = m 2 m 2 1 = σ2,... These are the cumulants. Paul Kirk 19 of 22

27 1.9 Moments and cumulants The moments can be calculated from the derivatives of the characteristic function, evaluated at s = 0. We can see this by expanding the characteristic function as a Taylor series: χ(s) = χ (n) (0)s n n=0 where χ (n) (s) is the n-th derivative of χ(s). But we also have: χ(s) = e isx (isx ) n i n X n s n = = n! n! n=0 n! n=0 Equating the 2 expressions, we get: X n = χ(n) (0) i n Paul Kirk 20 of 22

28 1.9 Moments and cumulants Cumulants The n-th order cumulant of X, is the n-th derivative of the log of the characteristic function. For independent random variables, X and Y, if Z = X + Y then the n-th cumulant of Z is the sum of the n-th cumulants of X and Y. The Gaussian distribution is also the only absolutely continuous distribution all of whose cumulants beyond the first two (i.e. other than the mean and variance) are zero. Paul Kirk 21 of 22

29 1.10 The multivariate Gaussian Let x = [x!,..., x N ], then the general form of the Gaussian pdf is: 1 P(x) = ( (2π) N det(σ) exp 1 ) 2 (x µ) Σ 1 (x µ), where µ is the mean vector and Σ is the covariance matrix. All higher moments of a Gaussian can be written in terms of the means and covariances. Defining X X X, for a 1-dimensional Gaussian we have: X 2n = X 2n 1 = 0 (2n 1)!(V [X ])n 2 n 1 (n 1)! Paul Kirk 22 of 22