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,

• 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

Sum of Normally Distributed RV

0.00

0.10

0.20

0.30

0.40

0.50

-4 -2 0 2 4

x~N[.5,1]

y~N[1,1]

x+y~N[1.5,2]

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

2.5% of the

distribution

2.5% of the

distribution

falls in this region

95% of the time

/

X

s n

Critical valuesCritical 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

Distribution of Stock Returns

S&P 500

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.80 -0.20 0.40 1.00 1.60 2.20 more

Distribution of Stock Returns

TSE 200

0

0.05

0.1

0.15

0.2

-0.80 0.20 1.20 2.20

Distribution of Stock Returns

DAX

0

0.020.04

0.060.08

0.10.12

0.14

-0.80 0.20 1.20 2.20

Linear Regression (Harvey 1989)

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

O-54 J-68 F-82 O-95 J-09

Growth

Spread

Harvey 1989

Growth

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

Spread

GNP Growth

Harvey 1989

Growth

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

Spread

GNP Growth

Regression Line:

1: 5 5( )t t t tGrowth a b Spread u

Regression

• Minimize the squared residuals:

Regression in Matrix Form

• Regression equation:

• Minimize the squared residuals: