Prediction

Description:

• A random variable $X$ is observed, and then an attempt is made to predict the value of a random variable $Y$ .
• Let $g(X)$ denote the predictor
  • that is, if $X$ is observed to equal $x$
  • then $g(x)$ is our prediction for the value of $Y$ .
  • We want to choose $g$ so that $g(X)$ tends to be close to $Y$ .
• One possible way is to choose $g$ to minimize $E[(Y-g(X){)}^2]$
  • The best possible preditor is $g(X)=E(Y|X)$
• Proposition:
  • $E[(Y-g(X){)}^2]\ge E[(Y-E[Y|X]{)}^2]$
  • $Var(X)=0$ then $P(X=E[X])=1$

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