Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 1 Physical Fluctuomatics Applied Stochastic Process 3rd Random variable, probability distribution and probability density function Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 2 Probability a.Event and Probability b.Joint Probability and Conditional Probability c.Bayes Formula, Prior Probability and Posterior Probability d.Discrete Random Variable and Probability Distribution e.Continuous Random Variable and Probability Density Function f.Average, Variance and Covariance g.Uniform Distribution h.Gauss Distribution Last Talk Present Talk

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 3 Probability and Random Variable We introduce a one to one mapping X(A) from every events A to a mutual different real number. The mapping X(A) is referred to as Random Variable of A. The random variable X(A) is often denoted by just the notation X. Probability of the event X=x that the random variable X takes a real number x is denoted by Pr{X=x}. Here, x is referred to as the state of the random variable X ． The set of all the possible states is referred to as State Space. If events X=x and X=x’ are exclusive of each other, the states x and x’ are excusive of each other.

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 4 Discrete Random Variable and Continuous Random Variable Discrete Random Variable: Random Variable in Discrete State Space Example:{x 1,x 2,…,x M } Continuous Random Variable: Random Variable in Continuous State Space Example ： (−∞,+∞)

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 5 Discrete Random Variable and Probability Distribution If all the probabilities for the events X=x 1, X=x 2,…, X=x M are expressed in terms of a function P(x) as follows: the function P(x) and the variable x is referred to as Probability Distribution and State Variable, respectively. Random VariableState Variable State Let us suppose that the sample  is expressed by Ω=A 1 ∪ A 2 ∪ … ∪ A M where every pair of events A i and A j are exclusive of each other. We introduce a one to one mapping X:A i  x i (i=1,2,…,M).

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 6 Discrete Random Variable and Probability Distribution Probability distributions have the following properties: Normalization Condition

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 7 Average and Variance Average of Random Variable X : μ Variance of Random Variable X: σ 2  ： Standard Deviation

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 8 Discrete Random Variable and Joint Probability Distribution If the joint probability Pr{(X=x)∩(Y=y)}= Pr{X=x,Y=y} is expressed in terms of a function P(x,y) as follows: P(x,y) is referred to as Joint Probability Distribution. Probability Vector State Vector

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 9 Discrete Random Variable and Marginal Probability Distribution Marginal Probability Distribution Summation over all the possible events in which every pair of events are exclusive of each other. Simplified Notation Normalization Condition Let us suppose that the sample  is expressed by Ω=A 1 ∪ A 2 ∪ … ∪ A M where every pair of events A i and A j are exclusive of each other. We introduce a one to one mapping X:A i  x i (i=1,2,…,M).

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 10 Discrete Random Variable and Marginal Probability Marginal Probability Distribution XY Z U Marginalize Marginal Probability of High Dimensional Probability Distribution

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 11 Independency of Discrete Random Variable If random variables X and Y are independent of each others, Joint Probability Distribution of Random Variables X and Y Probability Distribution of Random Variable X Probability Distrubution of Random Variable Y Marginal Probability Distribution of Random Varuiable Y

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 12 Covariance of Discrete Random Variables Covariance of Random Variables X and Y Covariance Matrix

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 13 Example of Probability Distribution a E[X] 0

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 14 Example of Joint Probability Distributions a Cov[X,Y] 0

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 15 Example of Conditional Probability Distribution Conditional Probability of Binary Symmetric Channel

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 16 Continuous Random Variable and Probability Density Function For a random variable X defined in the state space (−∞,+∞), the probability that the state x is in the interval (a,b) in expressed as Distribution Function Probability Density Function

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 17 Continuous Random Variable and Probability Density Function Normalization Condition

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 18 Average and Variance of Continuous Random Variable Average of Random Variable X Variance of Random Variable X

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 19 Continuous Random variables and Joint Probability Density Function 確率変数 X と Y の状態空間 (−∞,+∞) において 状態 x と y が区間 (a,b)×(c,d) にある確率 Joint Probability Density Function Normalization Condition For random variables X and Y defined in the state space (−∞,+∞), the probability that the state vector (x,y) is in the region (a,b)  (c,d) is expressed as

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 20 Continuous Random Variables and Marginal Probability Density Function Marginal Probability Density Function of Random Variable Y

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 21 Independency of Continuous Random Variables Random variables X and Y are independent of each other. Joint Probability Density Function of X and Y Probability Density Function of Y Marginal Probability Density Function Y Probability Density Function of X

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 22 Covariance of Continuous Random Variables Covariance of Random Variables X and Y Covariance Matrix

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 23 Uniform Distribution U(a,b) Probability Density Function of Uniform Distribution p(x)p(x) x 0 ab (b-a) -1

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 24 Gauss Distribution N(μ,σ 2 ) The average and the variance are derived by means of Gauss Integral Formula Probability Density Function of Gauss Distribution with average μ and variance σ 2 p(x)p(x) μ x 0

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 25 Multi-Dimensional Gauss Distribution by using the following d -dimentional Gauss integral formula For a positive definite real symmetric matrix C, two- Dimensional Gaussian Distribution is defined by The covariance matrix is given in terms of the matrix C as follows:

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 26 Law of Large Numbers Let us suppose that random variables X 1,X 2,...,X n are identical and mutual independent random variables with average  Then we have Central Limit Theorem tends to the Gauss distribution with average  and variance  2 /n as n  + . We consider a sequence of independent, identical distributed random variables, {X 1,X 2,...,X n }, with average  and variance    Then the distribution of the random variable

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 27 Summary a.Event and Probability b.Joint Probability and Conditional Probability c.Bayes Formula, Prior Probability and Posterior Probability d.Discrete Random Variable and Probability Distribution e.Continuous Random Variable and Probability Density Function f.Average, Variance and Covariance g.Uniform Distribution h.Gauss Distribution Last Talk Present Talk

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 28 Practice 3-1 Let us suppose that a random variable X takes binary values  1 and the probability distribution is given by Derive the expression of average E[X] and variance V[X] and draw their graphs by using your personal computer.

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 29 Practice 3-2 Derive the expressions of Marginal Probability Destribution of X, P(X), and the covariance of X and Y, Cov[X,Y]. Let us suppose that random variables X and Y take binary values  1 and the joint probability distribution is given by

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 30 Practice 3-3 Show that it is rewritten as Hint cosh(c) is an even function for any real number c. Let us suppose that random variables X and Y take binary values  1 and the conditional probability distribution is given by

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 31 Practice 3-4 Prove the Gauss integral formula: Hint

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 32 Practice 3-5 Prove that the average E[X] and the variance V[X] are given by Let us suppose that a continuous random variable X takes any real number and its probability density function is given by Draw the graphs of p(x) for μ=0, σ=10, 20, 40 by using your personal computer.

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 33 Practice 3-6 Make a program for generating random numbers of uniform distribution U(0,1). Draw histgrams for N generated random numbers for N=10, 20, 50, 100 and In the C language, you can use the function rand() that generate one of values 0,1,2,…,randmax, randomly. Here, randmax is the maximum value of outputs of rand().

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 34 Practice 3-7 Make a program that generates random numbers of Gauss distribution with average  and variance σ 2. Draw histgrams for N generated random numbers for N=10, 20, 50, 100 and For n random numbers x 1,x 2,…,x n generated by any probability distribution, (x 1 +x 2 +…+x n )/n tends to the Gauss distribution with average m and variance σ 2 /n for sufficient large n. [Central Limit Theorem] First we have to generate twelve uniform random numbers x 1,x 2,…,x 12 in the interval [0,1]. Gauss random number with average 0 and variaince 1 σ ξ +μ generate Gauss random numbers with average μ and variance σ 2 Hint:

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 35 Practice 3-8 For any positive integer d and d  d positive definite real symmetric matrix C, prove the following d- dimensional Gauss integral formulas: By using eigenvalues λ i and their corresponding eigenvectors (i=1,2,…,d) of the matrix C, we have Hint:

Physics Fluctuomatics/Applied Stochastic Process (Tohoku University) 36 Practice 3-9 Prove that the average vector is and the covariance matrix is C. We consider continuous random variables X 1,X 2,…,,X d. The joint probability density function is given by