6.041 / 6.431 9. Multiple Continuous Random Variables

LECTURE 9 • Readings: Sections 3.4-3.5 Outline • PDF review • Multiple random variables – conditioning – independence • Examples Summary of concepts pX(x) fX (x) FX (x) ! x xpX(x) E[X] " xfX(x) dx var(X) pX,Y (x, y) fX,Y (x, y) pX|A(x) fX|A(x) pX|Y (x | y) fX|Y (x | y) Continuous r.v.’s and pdf’s Sample Space x fX(x) Event {a < X < b } a b P(a ! X ! b)= "b a fX (x) dx • P(x ! X ! x + !) " fX(x) · ! • E[g(X)] = "# −# g(x)fX (x) dx Joint PDF fX,Y (x, y) P((X, Y ) % S)= "" S fX,Y (x, y) dx dy • Interpretation: P(x ! X ! x+!,y ! Y ! y+!) " fX,Y (x, y)·!2 • Expectations: E[g(X, Y )] = "# −# "# −# g(x, y)fX,Y (x, y) dx dy • From the joint to the marginal: fX(x) · ! " P(x ! X ! x + !)= • X and Y are called independent if fX,Y (x, y)= fX (x)fY (y) Buffon’s needle • Parallel lines at distance d Needle of length " (assume " 0 • For given y, conditional PDF is a (normalized) “section” of the joint PDF • If independent, fX,Y = fX fY , we obtain fX|Y (x|y)= fX(x) Stick-breaking example • Break a stick of length " twice: break at X: uniform in [0, 1]; break again at Y , uniform in [0,X] x fY|X (y | x) fX(x) L y fX,Y (x, y)= fX(x)fY |X (y | x)= on the set: x y L L E[Y | X = x]= " yfY |X(y | X = x) dy = fX,Y (x, y)= 1 "x , 0 ! y ! x ! " x y L L fY (y)= " fX,Y (x, y) dx = " " y 1 "x dx = 1 " log " y , 0 ! y ! " E[Y ]= " " 0 yfY (y) dy = " " 0 y 1 " log " y dy = " 4 Joint, marginal, and conditional densities.Image by MIT OpenCourseWare, adapted fromProbability, by J. Pittman, 1999.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.