6.041 / 6.431 11. Derived Distributions; Convolution; Covariance

LECTURE 11 Derived distributions; convolution; covariance and correlation • Readings: Finish Section 4.1; Section 4.2 Example x1 y 1 f X ,Y(y,x)=1 Find the PDF of Z = g(X, Y )= Y /X FZ(z)= z ! 1 FZ(z)= z " 1 A general formula • Let Y = g(X) g strictly monotonic. x g(x) y [x, x+d] [y, y+?] slope dg dx (x) • Event x ! X ! x + ! is the same as g(x) ! Y ! g(x + !) or (approximately) g(x) ! Y ! g(x)+ !|(dg/dx)(x)| • Hence, !fX(x) = !fY (y) !!!! dg dx(x) !!!! where y = g(x) The distribution of X + Y • W = X + Y ; X, Y independent x y . . . . . . (3,0) (2,1) (1,2) (0,3) pW (w) = P(X + Y = w) = " x P(X = x)P(Y = w − x) = " x pX(x)pY (w − x) • Mechanics: – Put the pmf’s on top of each other – Flip the pmf of Y – Shift the flipped pmf by w (to the right if w> 0) – Cross-multiply and add The continuous case • W = X + Y ; X, Y independent x y x + y = w w w • fW |X(w | x)= fY (w − x) • fW,X(w, x)= fX(x)fW |X(w | x) = fX(x)fY (w − x) • fW (w)= # $ −$ fX(x)fY (w − x) dx Two independent normal r.v.s •X % N(µx, "2 x), Y % N(µy, "2 y ), independent fX,Y (x, y)= fX(x)fY (y) = 1 2#"x"y exp $ − (x − µx)2 2"2 x − (y − µy)2 2"2 y % •PDF is constant on the ellipse where (x − µx)2 2"2 x + (y − µy)2 2"2 y is constant •Ellipse is a circle when "x = "y The sum of independent normal r.v.’s •X % N(0, "2 x), Y % N(0, "2 y ), independent •Let W = X + Y fW (w)= # $ −$ fX(x)fY (w − x) dx = 1 2#"x"y # $ −$ e−x2/2"2 x e−(w−x)2/2"2 y dx (algebra) = ce−$w2 •Conclusion: W is normal – mean=0, variance="2 x + "2 y – same argument for nonzero mean case Covariance •cov(X, Y )= E &(X − E[X]) ·(Y − E[Y ])' •Zero-mean case: cov(X, Y )= E [XY ] x . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y x •cov(X, Y )= E[XY ] − E[X]E[Y ] •var ( ) n" i=1 Xi * + = n" i=1 var(Xi)+2 " (i,j):i&=j cov(Xi,Xj) •independent ' cov(X, Y )=0 (converse is not true) Correlation coefficient •Dimensionless version of covariance: % = E ,(X − E[X]) "X · (Y − E[Y ]) "Y - = cov(X, Y ) "X"Y •−1 ! % ! 1 •|%|=1 ( (X − E[X]) = c(Y − E[Y ]) (linearly related) •Independent ' % =0 (converse is not true) MIT OpenCourseWare http://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.