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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. • 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, • Operationally: two variables are indepndent iff their joint pdf can be “separated” for any x and y:
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
Conditional Distributions (Discrete) • For any two events, E and F, • Conditional pdf: • Examples Ch 6, 4a, 4b
Conditional Distributions (Discrete) • Conditional cdf:
Conditional Distributions (Discrete) • Example: what is the probability that the TSX is up, conditional on the S&P500 being up?
Conditional Distributions (Continous) • Conditional pdf: • Conditional cdf: • Example 5b
Conditional Distributions (Continous) • Example: what is the probability that the TSX is up, conditional on the S&P500 being up 3%?
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:
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?
Joint PDF of Functions of R.V. • Example 7a – uniform and normal cases
Estimation • Given limited data we make educated guesses about the true parameters • Estimation of the mean • Estimation of the variance • Random sample
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
Distribution of the Sample Mean • The sample mean follows a t-distribution:
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
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
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
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
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
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
Harvey 1989 GNP Growth Spread
Harvey 1989 Regression Line: GNP Growth Spread
Regression • Minimize the squared residuals:
Regression in Matrix Form • Regression equation: • Minimize the squared residuals: