Multivariate random variables Flashcards
Conditional distribution properties and context in bivariate variables?
P(AlB) = P(AnB) / P(B)
=P ylx (ylx)
COV (X,Y)?
E(XY) - E(X)E(Y)
Properties of covariance?
-COV(X,X) = Var (X)
-COV(a,X) = 0
COV (aX+b, cY+d) = ac COV (X,Y)
CORR (X,Y)?
COV (X,Y) / sd(X)sd(Y)
When are x and y said to be uncorrelated?
-When COV(X,Y) = 0
When CORR (X,Y) >0
-positively correlated
-with y=aX+b with a>0
When CORR (X,Y) <0
-negatively correlated
-with y=aX+b with a<0
E(XY)?
-we just workout possibilities multiply to make xy then multiply by xy like normal
Sample covariance?
-1/n-1 sum of (x-x̄)(y-ȳ)
-where n is the number of inputs at x and y
-and x-x̄ is the deviation from sample mean
Sample correlation
sum of(x-x̄)(y-ȳ) / √(sum of squared x deviation)(sum of squared y deviation)
When are X and Y independent?
when Px,y (x,y) = Px(x)Py(y)
What does independence bring?
COV(X,Y) = CORR(X,Y) =0
-does not prove independence
- as even with correlation of zero, dependence may be non linear
E(X+Y) and E(X-Y)
-E(X)+E(Y)
-E(X)-E(Y)
when a1=a2=1
and b=0
How do you work out probability function for random variables?
- product
Var(X+Y) and VAR(X-Y)?
-Var(X) +Var(Y) +2 x Cov(X,Y)
-Var(X)+Var(Y)- 2xCov (X,Y)
if independent covariance is zero
If random variables are independent with same probability, how do we model with binomial distribution?
X-Bin(sum of n , Π)
If random variables are independent with same probability, how do we model with Poisson distribution?
X-Poi(sum of λ)
How do we model two normal distributions independent of eachtoher
-E(X)+E(Y) if +
-E(X)-E(Y) if -
-Var(X) +Var(Y) for + and -
