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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 Distributions of R. V.

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

  • Example Ch 6, 1c, 1d


Independence
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 v1
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
Conditional Distributions (Discrete)

  • For any two events, E and F,

  • Conditional pdf:

  • Examples Ch 6, 4a, 4b



Conditional distributions discrete2
Conditional Distributions (Discrete)

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


Conditional distributions continous
Conditional Distributions (Continous)

  • Conditional pdf:

  • Conditional cdf:

  • Example 5b


Conditional distributions continous1
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 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 v1
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 v2
Joint PDF of Functions of R.V.

  • Example 7a – uniform and normal cases


Estimation
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 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
Distribution of the Sample Mean

  • The sample mean follows a t-distribution:


Confidence intervals
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
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 intervals1
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
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
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
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
Harvey 1989

GNP Growth

Spread


Harvey 19891
Harvey 1989

Regression Line:

GNP Growth

Spread


Regression
Regression

  • Minimize the squared residuals:


Regression in matrix form
Regression in Matrix Form

  • Regression equation:

  • Minimize the squared residuals:


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