Joint distributions of r v
This presentation is the property of its rightful owner.
Sponsored Links
1 / 29

Joint Distributions of R. V. PowerPoint PPT Presentation


  • 88 Views
  • Uploaded on
  • Presentation posted in: General

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 ,

Download Presentation

Joint Distributions of R. V.

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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


Sum of normally distributed rv

Sum of Normally Distributed RV


Conditional distributions discrete

Conditional Distributions (Discrete)

  • For any two events, E and F,

  • Conditional pdf:

  • Examples Ch 6, 4a, 4b


Conditional distributions discrete1

Conditional Distributions (Discrete)

  • Conditional cdf:


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


Distribution of stock returns

Distribution of Stock Returns


Distribution of stock returns1

Distribution of Stock Returns


Distribution of stock returns2

Distribution of Stock Returns


Linear regression harvey 1989

Linear Regression (Harvey 1989)


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:


  • Login