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Joint Distributions of R. V.

Joint Distributions of R. V. Joint probability distribution function: f ( x,y ) = P ( X=x, Y=y ) Example Ch 6, 1c, 1d. Independence. Two variables are independent if, for any two sets of real numbers A and B ,

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Joint Distributions of R. V.

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

  2. 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:

  3. 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

  4. Sum of Normally Distributed RV

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

  6. Conditional Distributions (Discrete) • Conditional cdf:

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

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

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

  10. 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:

  11. Joint PDF of Functions of R.V. • Under the conditions on previous slide, • 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?

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

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

  14. Population vs. Sample • Population parameter describes the true characteristics of the whole population • Sample parameter describes characteristics of the sample • Statistics is all about using sample parameters to make inferences about the population parameters

  15. Distribution of the Sample Mean • The sample mean follows a t-distribution:

  16. Confidence Intervals • We can estimate the mean, but we’d like to know how accurate our estimate is • We’d like to put upper and lower bounds on our estimate • We might need to know whether the true mean is above certain value, e.g. zero

  17. Constructing Confidence Intervals • We already know the distribution of our estimate of the mean • To construct a 95% confidence interval, for instance, just find the values that contain 95% of the distribution

  18. Constructing Confidence Intervals falls in this region 95% of the time 2.5% of the distribution 2.5% of the distribution Critical values Critical values

  19. Confidence Intervals and Hypothesis Testing • The critical values are available from a table or in Matlab >> tinv(.975, n-1) • If the confidence interval includes zero, then the sample mean is not statistically different from the population mean we are testing • One-sided vs. two-sided tests

  20. Example • Are the returns on the S&P 500 significantly above zero? • Sample mean = .23 • Sample standard deviation = .59 • Sample size = 128 • Compute the test: • At 95% the critical value is 1.98 • Therefore, we reject that the returns are zero

  21. Distribution of S&P500 Returns • The direct use of historical data requires the following assumptions: • The true distribution of returns is constant through time and will not change in the future • Each period represents an independent draw from this distribution

  22. Distribution of Stock Returns

  23. Distribution of Stock Returns

  24. Distribution of Stock Returns

  25. Linear Regression (Harvey 1989)

  26. Harvey 1989 GNP Growth Spread

  27. Harvey 1989 Regression Line: GNP Growth Spread

  28. Regression • Minimize the squared residuals:

  29. Regression in Matrix Form • Regression equation: • Minimize the squared residuals:

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