Some Basic Concepts of Probability and Information Theory: Pt. 2

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1 Some Basic Concepts of Probability and Information Theory: Pt. 2 PHYS 476Q - Southern Illinois University January 22, 2018 PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

2 Review: Random Variables For a discrete probability space (Ω, p), a real-valued discrete random variable is a function X : Ω X R. Associated with every random variable is a probability space (X, p X ), where p X is the probability distribution given by p X (x) = p(x 1 (x)). We say that X = x with probability p X (x). One use of random variables is to take non-mathematical objects (such as coin flips, seating arrangements, etc. ) and represent them with mathematical objects like numbers. In practice we do not need to have a specific experiment in mind when working with random variables. PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

3 Review: Random Variables Example An example we will keep in mind is the random variable X on set X = {0, 1, 2, 3} with distribution p X (0) = 1/8 p X (1) = 3/8, p X (2) = 3/8 p X (3) = 1/8. For real-valued random variables, we can consider events characterized by algebraic relationships such as X a, X [0, 1], X 2 = 4, etc. Example With X given in teh above example, let E be the event that X 2. Then Pr{X 2} := p X (E) = x 2 p X (x) = = 1 2. PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

4 Chebyshev s Inequality Theorem Let X be a random variable with expectation value E[X ] and non-zero variance σ 2 (X ). For any κ > 0, Pr{ X E[X ] κσ} < 1 κ 2. Proof PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

5 Functions of Random Variables If f : X X R, then we can define a new random variable X := f (X ) whose sample space is X and who takes on values x X with probability p X (x) = p X (f 1 (x)). Example Let f : {0, 1, 2, 3} {0, 1} be the function such that f (x) = 0 if x is even and f (1) = 1 if x is odd. Set Y = f (X ). Then Y is the random variable ranging over the set Y = {0, 1} with distribution p Y (0) = 1 p Y (1) = PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

6 The Entropy of a Random Variable Definition For a distribution p X over X, its self-information is the function J : X (0, + ] given by J(x) = log p X (x). Roughly speaking, the self-information quantifies how surprised you would be if outcome x occurred in an experiment described by random variable X. Graph of Self-Information PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

7 The Entropy of a Random Variable Definition The Shannon entropy of a random variable X is the expectation value of its self information: H(X ) = E[J(X )] = x X p X (x) log p X (x). Example A binary random variable is a random variable ranging over two values: PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

8 Joint Random Variables: A Motivating Example Consider again the variable X with distribution p X (0) = 1/8 p X (1) = 3/8, p X (2) = 3/8 p X (3) = 1/8. Suppose we use the value of X to generate a new variable Y according to the following prescription: 1 If X = 0, 3, then set Y = 0. 2 If X = 1, 2, then flip a fair coin and set Y = { 0 if heads 1 if tails. PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

9 Joint Random Variables: A Motivating Example What is the joint sample space for variables X and Y and the distribution over outcomes? What is the distribution over outcomes for just the variable Y? PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 2 22, / 14

10 Joint Random Variables For two sets X and Y, a joint distribution is a probability distribution p XY over the product set X Y. The pair XY are called joint random variables with probability space (X Y, p XY ). The marginal or reduced distributions of p XY are distributions p X and p Y ranging over sets X and Y respectively and given by the formulas p X (x) = y Y p XY (x, y) p Y (y) = x X p XY (x, y). PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 222, / 14

11 Joint Random Variables Two random variables X and Y are called independent or uncorrelated if p XY (x, y) = p X (x) p Y (y) (x, y) X Y. Otherwise the variables are correlated. Example Are the variables X and Y defined in the previous example independent? PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 222, / 14

12 Joint Random Variables Example For uncorrelated random variables X and Y, what is the expectation value and variance of their sum X + Y? PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 222, / 14

13 Conditional Probability and Bayes Rule Definition For random variables X and Y with joint distribution p XY, whenever p Y (y) 0, the conditional distribution of X given Y = y is p X Y =y (x) := p XY (x, y). p Y (y) Likewise, whenever p X (x) 0, the conditional distribution of Y given X = x is p Y X =x (y) := p XY (x, y). p X (x) Bayes Rule p Y X =x (y) = p Y X =x(y)p X (x). p Y (y) PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 222, / 14

14 Conditional Probability and Bayes Rule Example Suppose two balls are placed in a hat, each can be either red, black, or green with probability 1/3. You draw one ball and see that it is green. You place it back in the hat and draw another ball. (a) What is the probability that your second draw is red? (b) What is the probability that your second draw is green? PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and Information Theory: January Pt. 222, / 14

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