By Forrest M., III Mims
This booklet contains standart program circuits and circuits designed through the writer
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Extra info for Engineer's Mini-Notebook: Optoelectronics Circuits
Example text
A linear function of a Gaussian random variable is also a Gaussian random variable. Now consider the sum of two independent Gaussian random variables, X and Y . We will calculate the PDF of X +Y by making use of the characteristic function. Because X and Y are independent, we find that: ϕX+Y (t) = ϕX (t)ϕY (t). In Chapter 7 we develop the tools to calculate the characteristic function of a Gaussian PDF. The characteristic function that corresponds to: fV (α) = √ 2 2 1 e−(α−µ) /(2σ ) 2πσ is: 2 ϕV (t) = e−jtµ e−t σ 2 /2 .
Let us calculate the expected value and the variance of Xi . 0196. The value we are interested in, the total number of times the bullseye was hit, is just: Y = X1 + · · · + X10000 . Let us calculate the expected value and the standard deviation of Y . 98 = 9800. The variance of Y is: E((Y − E(Y ))2 ) = E((X1 + · · · + X10000 − E(X1 ) − · · · − E(X10000 ))2 ) = E(((X1 − E(X1 )) + · · · + (X10000 − E(X10000 )))2 ) = E((X1 − E(X1 ))2 + · · · + (X10000 − E(X10000 ))2 (Xi − E(Xi ))(Xj − E(Xj ))) + i=j = E((X1 − E(X1 ))2 ) + · · · + E((X10000 − E(X10000 ))2 ) E((Xi − E(Xi ))(Xj − E(Xj ))).
The importance of this relation cannot be overemphasized. This result holds for discrete random variables too. For the proof, see Problem 12. 4 21 Correlation If two random variables X and Y are independent, then E(XY ) = E(X)E(Y ). If all that we know is that E(XY ) = E(X)E(Y ), then we say that random variables are uncorrelated. Thus, all independent random variables are uncorrelated, but uncorrelated random variables need not be independent. If E(XY ) = E(X)E(Y ), then X and Y are said to be correlated.