joint distributions of r. v
DESCRIPTION
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 , - PowerPoint PPT PresentationTRANSCRIPT
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: