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# Joint Distributions of R. V. - PowerPoint PPT Presentation

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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Presentation Transcript

• Joint probability distribution function: f(x,y) = P(X=x, Y=y)

• Example Ch 6, 1c, 1d

• 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:

• 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

• For any two events, E and F,

• Conditional pdf:

• Examples Ch 6, 4a, 4b

• Conditional cdf:

• Example: what is the probability that the TSX is up, conditional on the S&P500 being up?

• Conditional pdf:

• Conditional cdf:

• Example 5b

• Example: what is the probability that the TSX is up, conditional on the S&P500 being up 3%?

• = joint pdf of X1 and X2

• Equations and can be uniquely solved for and given by:

and

• The functions and have continuous partial derivatives:

• 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?

• Example 7a – uniform and normal cases

• Given limited data we make educated guesses about the true parameters

• Estimation of the mean

• Estimation of the variance

• Random 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

• The sample mean follows a t-distribution:

• 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

• 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

falls in this region 95% of the time

2.5% of the distribution

2.5% of the distribution

Critical values

Critical values

• 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

• 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

• 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

GNP Growth

Regression Line:

GNP Growth