Archive for the ‘statistics’ Category
STAT: Multivariate
Distribution function for random variable
Density Function
Expectation
Independence
 Given independent X_{i}‘s such that
Multivariate Moment Generating Function
Independent X_{i}:
Conditional Density
Let
Covariance Matrix
 Let , then and
 Nonnegative Definite, i.e.,
Let , then  then
Standardization of random variable
Since is nonnegative definite, there exists a matrix A such that
In addition, if it is positive definite, then A is invertible. Now let
Then
 Z is standardized: a) Z_{i} are uncorrelated, b) E[Z_{i}] = 0; c) Var[Z_{i}] = 1
Another Standardization of random variable
Let us say

where
i.e.
Here too Z is standardized: a) but Z_{i} are not uncorrelated, b) E[Z_{i}] = 0; c) Var[Z_{i}] = 1
Actually, covariance matrix of Z is same as correlation matrix of X :
Change of variable – multivariate
Given:
 C be the smallest open set in so that
 Let is onetoone and continuously differentiable
 y has an inverse x. So that X = x(Y)
 X has probability density function f_{X}
Then probability density function of Y is given by:
Here is the Jacobian Matrix defined by
STAT: Bivariate
Distribution function for random variable
For
Density Function
Marginal Density
Expectation
Covariance
Variance
Pearson Correlation
Independence
Show that
1) given this condition implies independence (P(AB) = P(A)P(B)). Use E[1_{A}] = P(A)
2) given independence, this equality follows. Just follows from definition of E.
Uncorrelated
X_{1} and X_{2} are uncorrelated if Cov(X_{1}, X_{2}) = 0
 Independent implies Uncorrelated
 Uncorrelated does not necessarily imply Independent
Only for linear functions of the form
Independence requires it to hold for all functions not just linear functions.  Independent ⊂ Uncorrelated
 Example: Intraday stock returns are dependent (not independent) and correlated (not uncorrelated, Cov ≠ 0).
Interday Stock returns (from yesterday and today) are uncorrelated but not independent (still dependent).
Conditional Distribution
Conditional Density
Conditional Expectation
If X_{1} and X_{2} are independent,
Conditional Variance