Discrete Random Variable

1. Discrete Random Variable • Let X denote the return of the S&P 500 tomorrow, rounded to the nearest percent • what are the possibilities, i.e. 0, 1%, … • what is the probability of each of the above possibilities • Probability distribution function: f(x) = P(X=x)

2. Probability Distribution

3. Cumulative Distribution Function

4. Discrete Random Variable • Expectation • Variance

5. Which Distribution Has Higher Mean?

6. Which Distribution Has Higher Variance?

7. Expectation of a Function of a R.V. • Function g(X): • What is the expectation E(g(X))? • General result: • Example – call option on the S&P 500

8. Binomial Distribution • Bernoulli distribution • A r.v. X has two possible outcomes, 0 or 1 • Binomial distribution • Number of successes that occur in n trials • Example: Ch. 4, 6b

9. Poisson • A r.v. X takes on values 0, 1, 2, .... • Poisson distribution iffor some l > 0, • The Poisson r.v. is an approximation for binomial with l = np. • Example: how many days in a year will the S&P500 drop more than 1%? • Example 7b

10. Geometric • Independent trials with prob. of success p • How many trials until a success occurs? • What happens when n goes to infinity? • Example: how many days until we get a stock market drop of 2% or more?

11. Negative Binomial • Independent trials with prob. of success p • How many trials until r successes occur? • What happens when n goes to infinity? • Example: how many days until we get three stock market drops of 2% or more (not necessarily consecutive)?

12. Hyper-Geometric • Choose n balls out of N, without replacement • m white, N – m black • X = number of white balls selected • Example 8i • What happens if you choose the n balls with replacement?

13. Continuous Random Variable • Let X denote the return of the S&P 500 tomorrow, no rounding • what are the possibilities • what is the probability of each of the above possibilities • Probability density function:

14. Probability Density Function

15. Cumulative Density Function

16. Continuous Random Variable • Expectation • Variance • Example Ch5, 1a, 1b, 2a

17. Continuous Random Variable • For any real-valued function g and continuous r.v. X: • Example: payoff on a call option, 2b

18. Which Distribution Has Higher Mean?

19. Which Distribution Has Higher Mean?

20. Which Distribution Has Higher Variance?

21. Skewness

22. Kurtosis

23. Additional Sample Questions • Given a discrete probability function (pdf) (i.e., all possible outcomes and their probabilities), compute the mean and the variance • Given a graph of several discrete or continuous pdf, estimate which ones has the highest mean, variance, skewness, kurtosis • Given two random variables, guess whether they have positive or negative covariance and/or correlation

24. The Uniform Distribution Example 3b

25. The Normal Distribution

26. The Normal Distribution Example Ch 5, 4b, 4e

27. Properties of the Normal Distribution

28. Normal is an Approximation to Binomial • Sn = number of successes in n independent trials with individual prob. of success p. • The DeMoivre-Laplace limit theorem:

29. Normal is an Approximation to Binomial

30. Lognormal Distribution • What is the distribution of the S&P 500 index tomorrow? • If the return on the S&P500 is normally distributed, the index itself is lognormally distributed

31. Lognormal Distribution

32. Chi-squared Distribution • Sum of squared standard normal variables

33. F distribution • Ratio of two independent chi-squared variables with degrees of freedom n1 and n2

34. t distribution • Very important for hypothesis testing

35. Normal vs. t distribution

36. Exponential Distribution • PDF: • CDF: • Exercise:

37. Joint Distributions of R. V. • Joint probability distribution function: f(x,y) = P(X=x, Y=y) • Example Ch 6, 1c, 1d

38. Independence • Two variables are independent if, for any two sets of real numbers A and B, • Operationally: two variables are indepndent iff their joint pdf can be “separated” for any x and y:

39. Joint Distributions of R. V. • The expectation of a sum equals the sum of the expectations: • The variance of a sum is more complicated: • If independent, then the variance of a sum equals the sum of the variances

40. Sum of Normally Distributed RV

41. Additional Sample Questions • Find the distribution of a transformation of two or more normal random variables • By looking at a graph of a pdf, guess whether it is normal, log-normal, or t-distribution • What normally distributed random variables do you need to construct an F distribution with 3 and 5 degrees of freedom

42. Conditional Distributions (Discrete) • For any two events, E and F, • Conditional pdf: • Examples Ch 6, 4a, 4b

43. Conditional Distributions (Discrete) • Conditional cdf:

44. Conditional Distributions (Discrete) • Example: what is the probability that the TSX is up, conditional on the S&P500 being up?

45. Conditional Distributions (Continous) • Conditional pdf: • Conditional cdf: • Example 5b

46. Conditional Distributions (Continous) • Example: what is the probability that the TSX is up, conditional on the S&P500 being up 3%?

47. Joint PDF of Functions of R.V. • = joint pdf of X1 and X2 • Equations and can be uniquely solved for and given by: and • The functions and have continuous partial derivatives:

48. Joint PDF of Functions of R.V. • Under the conditions on previous slide, • Insert eq. 7.1, p275 • Example: You manage two portfolios of TSX and S&P500: • Portfolio 1: 50% in each • Portfolio 2: 10% TSX, 90% S&P 500 • What is the probability that both of those portfolios experience a loss tomorrow?

49. Joint PDF of Functions of R.V. • Example 7a – uniform and normal cases

50. Estimation • Given limited data we make educated guesses about the true parameters • Estimation of the mean • Estimation of the variance • Random sample