Functions of random Variables

Function of a random variable

• Consider operations on a random variable is X

  • this shifts/scales X
    • Assuming that X has a mean $\mu$ and varince $\sigma$
  • Let c be a constant in this operation

  • Addition and subtraction

    • we define this new random varible by adding c: $Y=X+c$
    • mean and varince of Y becomes
      • $\text{mean shifts by c: }E(Y)=E(x)+c=\mu +c$
      • the variance does not change $V(Y)=V(x)+o={\sigma }^2$

  • Multiplication and division

    • we define the random varible as $Y=cX$
    • mean and variance of Y becomes
      • mean is shifted by c $E(Y)=E(cX)=c\mu$
      • variance is scaled by c aswell $V(Y)=V(cX)=c^2{\sigma }^2$
        • variance scales with X by c

Linear Functions of random varibles

• Different cases when the random variables are not independent

  • consider $X_1\text{ and }{X}_2$ being not independent
    • $Y=X_1+X_2$
      • since mean is additive we know that
        • $E(Y)=E(X_1+X_2)=E(X_1)+E(X_2)={\mu }_1+{\mu }_2$
      • However since its variance and the formula is the square along the square product
        • $V(Y)=E(Y^2)-(E(Y){)}^2$
          • and since we know that $E(X_1+X_2)={\mu }_1+{\mu }_2$

General Linear function for Dependent random variables

• Generally we have n random variables $X_n$

  • X is actually vectors or column matrix with x values
  • this gives us data for all the data points
  • each X has its own mean $\mu$ and variance ${\sigma }^2$

  • Consider we have n+1 constants c

    • $Y=c_0+c_1{X}_1+c_2{X}_2\dots$
      • $⟹c_0+{\sum }_{i=1}^n{c}_i{X}_i$
    • The Mean is additive so its just the same as before where its scaled by c
    • However we must consider covariance $(Cov(X_i{X}_j))$
      • where i < j
      • general variance becomes
        • $V(Y)={\sigma }_y^2={\sum }_{i=1}^n(c_i{\sigma }_i{)}^2+2{\sum }_{i=1}^n{\sum }_{j>i}[c_i{c}_j]Cov(X_i{X}_j)$