Independent random variables Discrete random variables X and Y are independent if the value taken by one does not affect the value taken by the other. P(X=x and Y=y)=P(X=x)P(Y=y) P(X=x Y=y)=P(X=x), P(Y=y X=x)=P(Y=y) Linear combinations For independent X,Y and real constants a,b, E(aX+- bY)=aE(X)+- bE(Y)Var(aX+- bY)=a2Var(X)+b2Var(Y) If X_1,X_2,,X_n are n independent observations of X, then E(X_1+. s+X_n)= _i=1nE(X_i)=nE(X)Var(X_1+. s+X_n)= _i=1nVar(X_i)=nVar(X) The multiple aX scales the values of X by a, so the data scale changes. The sum X_1+. s+X_n combines n independent observations of X; the scale of each observation is unchanged.

Discrete random variables \(X\) and \(Y\) are independent if the value taken by one does not affect the value taken by the other.

Joint probability

\[P(X=x\mid Y=y)=P(X=x),\qquad P(Y=y\mid X=x)=P(Y=y)\]

Conditional form

For independent \(X,Y\) and real constants \(a,b\),

\[E(aX\pm bY)=aE(X)\pm bE(Y)\]\[\operatorname{Var}(aX\pm bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)\]

Two independent variables

If \(X_1,X_2,\ldots,X_n\) are \(n\) independent observations of \(X\), then

\[E(X_1+\cdots+X_n)=\sum_{i=1}^{n}E(X_i)=nE(X)\]\[\operatorname{Var}(X_1+\cdots+X_n)=\sum_{i=1}^{n}\operatorname{Var}(X_i)=n\operatorname{Var}(X)\]

Sum of independent observations

The multiple \(aX\) scales the values of \(X\) by \(a\), so the data scale changes. The sum \(X_1+\cdots+X_n\) combines \(n\) independent observations of \(X\); the scale of each observation is unchanged.

Independent Random Variables and Linear Combinations

Independent Random Variables and Linear Combinations