Statistics Notes Fundamental Concepts And Solved Examples

Jointly Distributed Random Variables1. DefinitionTwo random variables Xand Yare said to be jointly distributed if there exists a functiondescribing the probability of all combinations of their values. Discrete Case (Joint PMF):pX,Y (x, y ) = P(X =x, Y =y), Xx Xy pX,Y (x, y ) = 1Continuous Case (Joint PDF): fX,Y (x, y ) 0, Z Z fX,Y (x, y )dx dy = 12. Marginal DistributionsDiscrete: pX (x ) = Xy pX,Y (x, y ), pY(y ) = Xx pX,Y (x, y )Continuous: fX (x ) = Z fX,Y (x, y )dy, fY(y ) = Z fX,Y (x, y )dx3. Conditional DistributionsDiscrete:P(X =x|Y =y) = P(X =x, Y =y) P(Y =y) , P(Y =y|X =x) = P(X =x, Y =y) P(X =x)Continuous:fX |Y (x |y ) = fX,Y (x, y ) fY (y ) , fY|X (y |x ) = fX,Y (x, y ) fX (x )4. Conditional ExpectationE[X |Y =y] = (PxxP(X =x|Y =y), discreteRxf X|Y (x |y )dx, continuousProperties: E[E [X |Y ]] = E[X ] Linearity: E[aX +bY |Z ] = aE[X |Z ] + bE[Y |Z ]15. Covariance and CorrelationCov(X, Y) =E[(X E[X ])( YE[Y ])] = E[X Y ] E[X ]E [Y ] X Y