## Some Important Theorems

- A necessary and sufficient condition for X and Y to be independent is that

F(x,y) = H(x)*G(y) \forall x,y

;where F(.) is the joint d.f. and H(.) and G(,) are the marginal distribution function of X and Y respectively.

- A necessary and sufficient condition for the independence of the discrete variables X and Y is that

p_{ij}=p_{i0}*p_{0j} \quad \forall i,j

- The distribution function F(.) of an absolutely continuous two-dimensional random variable (X,Y) is uniquely determined by its probability-density function. Conversely, the probability density function is uniquely determined by the distribution function except perhaps for a set of Lebesgue measure zero.

- A necessary and sufficient condition for the independence of the absolutely continuous variable X and Y is that

f(x,y)=h(x)*g(y) \forall x,y

- If var(X) and var(Y) exists, then covers(X,Y) also exists and

[cov(X,Y)]^2 \leq var(X)var(Y)

- If either X=c almost everywhere or Y=d almost everywhere (c and d being constants), then

cov(X,Y)=0

- If U=a+bX and V=c+dY, and if cov(X,Y) exists, then cov(U,V) also exists and

cov(U,V) = bd * cov(X,Y)

- If \mu_X,\mu_Y as well as \mu'_{11} exist, then cov(X,Y) too exists, and

\mu'_{11} = \mu_X\mu_Y

- Suppose var(X) and var(Y) both exist, then var(X+Y) also exists and

var(X+Y) = var(X) +var(Y) + 2cov(X,Y)

- If X and Y are independent, then

E(Y/X=x)=E(Y) \quad and \quad var(X/Y=y)=var(X)

- If E(Y/X=x) exists for almost all values of X then,

E(Y) = E_X E_Y \left[ Y/X \right]

- If E(Y/X) and var(Y/X) exist for almost all values of X, then

var(Y) = E (var(Y/X)) + var (E(Y/X))

- The correlation coefficient necessarily satisfies the inequality

-1 \leq \rho \leq 1

- The correlation takes values -1 or 1 iff X and Y are linearly related almost everywhere.

- If U=a+bX and V=c+dY, then \rho_{UV} = \pm \rho_{XY} , the sign depending upon whether b and d are of the same sign or of opposite signs.

- If X and Y are independent random variables, and if \rho is defined, then

\rho=0

- If the regression of Y on X is linear and if \sigma_X^2,\sigma_Y^2,\rho exist, then the constants \alpha, \beta are given by

\alpha = \mu_Y - \rho \frac{\sigma_Y}{\sigma_X} \mu_X , \beta = \rho \frac{\sigma_Y}{\sigma_X}

- If the regression of Y on X is linear and var(Y/X=x) is algebraically independent of x, then

var(Y/X=x) = \sigma_Y^2 (1-\rho^2)

## Leave a Reply