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Next: Variance 11/17 Up: Expectations and Variances for Previous: Expectations and Variances for

Properties

Proposition 1:

\begin{displaymath}\mbox{For any} X \mbox{ and } Y
E(X+Y)=E(X)+E(Y)
\end{displaymath}

Proposition 2:
If X is a continuous random variable with probability density function f(x) then for any real valued function g:

\begin{displaymath}E[g(X)]=\int_{-\infty}^{\infty}
g(x)f(x)dx\end{displaymath}

Proposition 3:

E(aX+b)=aE(X)+b

Proposition 4:
Only for independent random variables do we have

\begin{displaymath}If\;\; X\; and \; Y\; independent\;
E(XY)=E(X)E(Y)
\end{displaymath}

Examples:

For the exponential random variable with parameter $\lambda$, I showed:

\begin{displaymath}E[X]=\frac{1}{\lambda}\end{displaymath}

For U a random uniform on [0,1], I showed: $E(U)=\frac{1}{2}$.


\begin{displaymath}Z \sim Normal(0,1) E(Z)=0\end{displaymath}


\begin{displaymath}X \sim Normal(\mu,\sigma^2), E(X)=\mu\end{displaymath}



Susan Holmes
1998-12-07