Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
EMIS 7300 SYSTEMS ANALYSIS METHODS FALL 2005 Dr. John Lipp Copyright © 2003-2005 Dr. John Lipp Session 2 Outline • Part 1: Correlation and Independence. • Part 2: Confidence Intervals. • Part 3: Hypothesis Testing. • Part 4: Linear Regression. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-2 Today’s Topics • Bivariate random variables – Statistical Independence. – Marginal random variables. – Conditional random variables. • Correlation and Covariance – Multivariate Distributions. – Random Vectors. – Correlation and Covariance Matrixes. • Transformations – Transformations of a random variable. – Transformations of a bivariate random variables. – Transformations of a multivariate random variables. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-3 Bivariate Data • A common experimental procedure is to control one variable (input) and measure another variable (output). • The values of the “input” variable are denoted xi and the values of the “output” variable as yi. • An xy-plot of the data points is referred to as a scatter diagram if the data (xi and/or yi) are random. • From the scatter diagram a general data trend may be observed that suggests an empirical model. • Fitting the data to this model is known as regression analysis. When the appropriate empirical model is a line then the procedure is called simple linear regression. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-4 Bivariate Data (cont.) n xi yi 1 9.5013 22.4030 2 2.3114 12.3828 3 6.0684 20.3993 4 4.8598 17.5673 5 8.9130 27.9870 6 7.6210 22.9163 7 4.5647 18.8927 8 0.1850 0.5602 9 8.2141 21.3490 10 4.4470 13.2672 11 6.1543 10.8923 12 7.9194 21.8680 13 9.2181 19.2104 14 7.3821 25.4247 15 1.7627 5.3050 EMIS7300 Fall 2005 30 25 20 15 10 5 0 0 1 2 3 Copyright 2003-2005 Dr. John Lipp 4 5 6 7 8 9 10 5-5 Bivariate Data (cont.) • The line fit equation is yˆ mˆ x bˆ where 1 n n y x 1 n y n x yi y xi x i i i i n i 1 i 1 mˆ n i 1 n i 1 2 n n 1 2 1 x2 x xi x i i n i 1 i 1 n i 1 n n 1 bˆ y mˆ x yi mˆ xi n i1 i 1 EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-6 Simple Linear Regression (cont.) • The slope of the linear regression is related to the sample correlation coefficient 1 n xi x yi y s n i1 r x mˆ sy 1 n x x 2 1 n y y 2 i i n n i1 i 1 • The calculation for r can be rewritten as n y x 1 n y n x i i i i n i 1 i 1 i1 r n 2 1 n 2 n 2 1 n 2 yi yi xi xi i1 n i1 i 1 n i1 EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-7 Simple Linear Regression (cont.) • r has no units. • The value of r is bounded by 1. – r=1 – 0<r1 – r=0 – 0 > r -1 – r = -1 EMIS7300 Fall 2005 the line fits the data perfectly. the line has a positive slope. there is no line fit. the line has a negative slope. the line fits the data perfectly. Copyright 2003-2005 Dr. John Lipp 5-8 Simple Linear Regression (cont.) 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 0 0 1 1 2 2 3 3 EMIS7300 Fall 2005 4 4 5 5 6 6 7 7 8 8 9 9 10 10 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Copyright 2003-2005 Dr. John Lipp 5-9 Bivariate Random Variables • Consider the case of two random variables X and Y. • The joint CDF is denoted FX,Y(x,y) = P(X x, Y y). • The joint PDF is defined via the joint CDF x y FX ,Y ( x, y ) f X ,Y (u, v) du dv where f X ,Y ( x, y ) FX ,Y ( x, y ) x y • Expected value E X ,Y g ( x, y ) g ( x, y ) f X ,Y ( x, y ) dx dy EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-10 Statistical Independence • X and Y are statistically independent if and only if, FX ,Y ( x, y) FX ( x) FY ( y) or f X ,Y ( x, y) f X ( x) fY ( y) • Statistical Independence has an effect on the expected value of separable functions of joint random variables E X ,Y g ( x)h( y ) g ( x ) h( y ) f X ,Y ( x, y ) dx dy g ( x ) h( y ) f g ( x) f X ( x) fY ( y ) dx dy X ( x)dx h( y ) fY ( y ) dy E X {g ( X )}EY {h(Y )} EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-11 Marginal Random Variables • It is often of interest to find the individual CDFs and PDFs when two random variables are not statistically independent. These are known as the marginal CDF and marginal PDF. • Marginal CDFs are straightforward, x FX ( x) F ( x,) f X ,Y (u, v) dv du FY ( y ) F (, y ) f X ,Y (u, v) du dv y • Marginal PDFs are found by “integrating out” y or x, f X ( x) EMIS7300 Fall 2005 f X ,Y ( x, y) dy fY ( y ) Copyright 2003-2005 Dr. John Lipp f X ,Y ( x, y) dx 5-12 Conditional Random Variables • Conditional CDFs and PDFs can be defined, FX ,Y ( x, y) FX |Y ( x | y) FY ( y) FY | X ( y | x) FX ( x) f X ,Y ( x, y) f X |Y ( x | y) fY ( y) fY | X ( y | x) f X ( x) • Rewriting the conditional PDF for X given Y f X ,Y ( x, y ) f X |Y ( x | y ) fY ( y ) f X ,Y ( x, y ) f X ,Y ( x, y )dx fY | X ( y | x) fY ( y ) fY | X ( y | x) fY ( y )dx This is just ____________________ for random variables! • A similar equation holds for Y given X. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-13 Marginal Random Variables (cont.) • Consistent results are obtained if X and Y are independent, 1 FX (x) = FX,Y (x,) = FX (x) FY () = FX (x) f X ( x) 1 f X ,Y ( x, y) dy f X ( x) fY ( y)dy f X ( x) • Find the marginal PDFs for fX,Y(x,y) = 2 when 0 < x < y < 1 and fX,Y(x,y) = 0 everywhere else. Are X and Y independent? EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-14 Conditional Random Variables (cont.) • The definitions of conditional CDFs and PDFs are consistent when X and Y are statistically independent FX ,Y ( x, y ) FX ( x) FY ( y ) FX |Y ( x | y ) FX ( x) FY ( y ) FY ( y ) FX ,Y ( x, y ) FX ( x) FY ( y ) FY | X ( y | x) FY ( y ) FX ( x) FX ( x) f X ,Y ( x, y ) f X ( x) fY ( y ) f X |Y ( x | y ) f X ( x) fY ( y ) fY ( y ) f X ,Y ( x, y ) f X ( x) fY ( y ) fY | X ( y | x) fY ( y ) f X ( x) f X ( x) EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-15 Bivariate Guassian Random Variables • Let X and Y be jointly Gaussian, but not necessarily independent, random variables. • The joint PDF is f X ,Y ( x, y ) y2 x x 2 2 xy x y x x y y x2 y y 2 1 2 1 2 x 2 y 2 xy e 2 2 x2 y2 (1 xy ) • Note: E X ,Y { X } x E X ,Y {Y } y 2 E X ,Y { X x } E X ,Y { X 2 } x x x2 2 E X ,Y {Y y } E X ,Y {Y 2 } y y y2 E X ,Y { X x Y y } E X ,Y { X Y } x y x y xy EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-16 Bivariate Guassian Random Variables (cont.) • The marginal PDF of X ~ N( x,x2) and Y ~ N( y, y2). EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-17 Bivariate Guassian Random Variables (cont.) • Consider the case that the Gaussian variables are uncorrelated, that is, xy = 0. The joint PDF is then x x 2 1 2 x2 1 e f X ,Y ( x, y ) e 2 2 2 x 2 y f X ( x) fY ( y ) y y 2 2 y2 • Thus, uncorrelated jointly Gaussian random variables are independent Gaussian random variables. This is a very important exception to the notion that uncorrelated random variables are not also independent random variables. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-18 Correlation and Covariance • The correlation between two joint random variables X and Y is defined as E{XY}. • The covariance is defined as cov(X,Y) = E{(X - x)(Y - y)} = E{XY} - x y = xy where x and y are the means of X and Y, respectively. • X and Y are uncorrelated if and only if cov(X,Y) = 0. An equivalent condition is X and Y are uncorrelated if and only if E{XY} = E{X}E{Y}. This is not the same as independence! • Two random variables X and Y are said to be orthogonal if and only if E{XY} = 0. Not the same as uncorrelated! EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-19 Correlation and Covariance (cont.) • Independent random variables are always uncorrelated x y cov(X,Y) = E{XY} - x y = E{X}E{Y} - x y= 0 The reverse is generally not true. Uncorrelated RV’s Independent RVs EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-20 Correlation and Covariance (cont.) • The correlation coefficient (normalized covariance) is xy E X x Y y xy 2 2 x y E X x E Y y – – – – The correlation coefficient is bounded, -1 xy +1. xy = 0 if X and Y are uncorrelated. xy = 1 means that X and Y are perfectly correlated. xy = -1 means that X and Y are perfectly anti-correlated. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-21 Correlation and Covariance (cont.) • Although covariance describes a linear relationship between variables (if it exists), it does not give an indication of nonlinear relationships between variables. y 0.04 0.02 0.04 0.04 0.04 0.02 0.05 0.05 0.05 0.05 0.20 0.05 0.05 0.02 0.04 0.04 0.05 0.05 0.04 0.02 0.04 x • The above distribution shows a clear relationship between the random variables X and Y, but the covariance is zero! EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-22 Multivariate Distributions • When more than two random variables are considered, the various distributions and densities are termed multivariate. – Joint CDF: FX1,X2,…Xn (x1,x2,…,xn) – Joint PDF: fX1,X2,…Xn (x1,x2,…,xn) – Conditional CDF: FX1| X 2 ,...,X n ( x1 | x2 ,..., xn ) FX1 , X 2,...,X n ( x1 , x2 ,..., xn ) FX 2 ,...,X n ( x2 ,..., xn ) – Conditional PDF: f X1| X 2 ,...,X n ( x1 | x2 ,..., xn ) EMIS7300 Fall 2005 f X1 , X 2,...,X n ( x1 , x2 ,..., xn ) f X 2 ,...,X n ( x2 ,..., xn ) Copyright 2003-2005 Dr. John Lipp 5-23 Multivariate Distributions (cont.) – Marginal PDF: f X 2 ,...,X n ( x2 ,..., xn ) f X , X ,...,X ( x1 , x2 ,..., xn )dx1 1 2 n – Expectation: E{g ( x1 ,..., xn )} ... g ( x1 ,..., xn ) f X ,...,X ( x1 ,..., xn ) dx1 dxn 1 n – Independence: f X1 , X 2 ,...,X n ( x1 , x2 ,..., xn ) f X1 ( x1 ) f X 2 ( x2 )... f X n ( xn ) EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-24 Random Vectors • Using vector notation is just as useful for random variables as it is in other engineering disciplines. • Consider the random vector • Define the “vector PDF” X1 X 2 X X3 X n f ( x ) f X1 , X 2 ,...,X n ( x1 , x2 ,..., xn ) X • The CDF, marginal, and conditionals are similar. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-25 Correlation Matrix • Let X be an 1N random vector and Y be a 1M random vector. Then the correlation matrix, Rxy E{ XY T } , is • E{ X 1Y1} E{ X 1Y2 } E{ X 1Y3 } E{ X 1YM } E{ X Y } E{ X Y } E{ X Y } E{ X Y } 2 1 2 2 2 3 2 M T E{ XY } E{ X 3Y1} E{ X 3Y2 } E{ X 3Y3 } E{ X 3YM } E{ X N Y1} E{ X N Y2 } E{ X N Y3 } E{ X N YM } T Rx E{ XX } is known as the autocorrelation matrix. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-26 Covariance Matrix • Let X be an 1N random vector and Y be a 1M random vector. Then the covariance matrix is T C xy E( X x )(Y y ) x1 , y1 x1 , y2 x1 , y3 x1 , yM x2 , y2 x2 , y3 x2 , yM x2 , y1 x3 , y1 x3 , y2 x3 , y3 x3 , yM xN , y1 xN , y2 xN , y3 xN , yM where x and y are the vector means of X and Y , respectively. • It is often more useful or more natural to write T T C xy E( X x )(Y y ) E{ XY } x Ty Rxy x Ty EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-27 Covariance Matrix • Let X be an 1N random vector and Y be a 1M random vector. Then the covariance matrix is T C xy E( X x )(Y y ) cov( X 1 , Y1 ) cov( X 1 , Y2 ) cov( X 1 , Y3 ) cov( X 1 , YM ) cov( X , Y ) cov( X , Y ) cov( X , Y ) cov( X , Y ) 2 1 2 2 2 3 2 M cov( X 3 , Y1 ) cov( X 3 , Y2 ) cov( X 3 , Y3 ) cov( X 3 , YM ) cov( X N , Y1 ) cov( X N , Y2 ) cov( X N , Y3 ) cov( X N , YM ) where x and y are the vector means of X and Y , respectively. • It is often more useful or more natural to write T T C xy E( X x )(Y y ) E{ XY } x Ty Rxy x Ty EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-28 Covariance Matrix (cont.) • More interesting is the autocovariance matrix, 12 1 2 12 C x 1 3 13 1 N 1N 2 1 21 22 2 3 23 3 1 31 3 2 32 32 2 N 2 N 3 N 3 N N 1 N1 N 2 N1 N 3 N1 N2 • The autocovariance matrix is symmetric because ij = ji . • It is often more useful or more natural to write T T T T C x E ( X x )( X x ) E{ XX } x x Rx x x EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-29 Covariance Matrix (cont.) • Autocovariance matrix for uncorrelated random variables (ij = 0). 12 0 0 2 0 0 2 Cx 0 0 32 0 0 0 • Covariance matrix for perfectly correlated random variables (ij = 1). 1 2 C x 3 1 2 3 N N EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 0 0 0 N2 5-30 Covariance Matrix (cont.) • Consider a random variable Y which is the weighted sum of N independent random variables Xi, …, XN N T Y w1 X 1 w2 X 2 wN X N wi X i w X i 1 • The mean of Y is straight forward N T T y E{Y } E{w X } w x wi xi i 1 • The variance is also straight forward T 2 T 2 2 2 2 y E{Y } y E w X w x T T T T E{w X X w} w x x w EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-31 Covariance Matrix (cont.) • If the Xi are uncorrelated with different variances, then 12 0 0 0 2 0 0 0 2 N T 2 2 2 2 y w 0 0 3 0 w wi i i 1 0 0 0 N2 • If the Xi are correlated with different variances, then 2 1 1 2 2 2 N y2 wT 3 1 2 3 N w wT 3 wi i i1 N N EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-32 Covariance Matrix (cont.) • Let Y and b be M 1 vectors, A be an M N matrix, and X be an N 1 vector, then Y = AX + b has the statistics y A x b C y ACx AT • Usually it is easy to generate X as uncorrelated random variables with unit variances (Cx = identity matrix). • To generate Y with a desired autocovariance find the “square root” of Cy =AAT using eigenvector decomposition 12 0 0 1 0 0 2 0 0 0 0 2 2 U T A U C y UDU T U 0 0 2 0 0 0 0 0 0 N N EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-33 Covariance Matrix (cont.) • Covariance matrix for uncorrelated variables. • Covariance matrix after rotation rotation for uncorrelated! • How compute a sample correlation / covariance. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-34 Gaussian Vector • Let the elements of the random vector X be mutually Gaussian. The PDF in vector notation is 1 ( x x )T 1 ( x x ) 1 2 f X ( x ) e (2 ) N | | Determinant of where x is the mean and is the autocovariance matrix of X . • If the elements X are independent / uncorrelated (equivalent for Guassian only!) the inverse is trivial 0 1 0 2 1 EMIS7300 Fall 2005 0 22 0 1 12 0 0 2 0 0 2 2 N 0 0 Copyright 2003-2005 Dr. John Lipp 0 0 2 N 5-35 Bivariate Guassian Random Variables (cont.) • Consider the case that the Gaussian variables are uncorrelated, that is, xy = 0. The joint PDF is then x x 2 1 2 x2 1 e f X ,Y ( x, y ) e 2 2 2 x 2 y f X ( x) fY ( y ) y y 2 2 y2 • Thus, uncorrelated jointly Gaussian random variables are independent Gaussian random variables. This is a very important exception to the notion that uncorrelated random variables are not also independent random variables. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-36 • Transformation of RVs • Use inverse transformation to show Z = X+Y thing is convolution. • Use inverse transformation to show how to use uniform for generating other RVs. • See 232 in Papuolis book. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-37 Transformations of Random Variables • Many of the continuous random variables from the previous session where defined as non-linear functions of other random variables, e.g., a chi-square random variable is the result from squaring a zero-mean Gaussian random variable. • Here is how NOT to transform a random xvariable 1 – Let X ~ exponential, i.e., f X ( x) e , x 0 . – Define Y X and substitute X = Y 2 into fX(x), fY ( y ) f X ( y ) 2 1 e y2 , y0 – But Y should be Rayleigh, fY ( y ) EMIS7300 Fall 2005 y Copyright 2003-2005 Dr. John Lipp e y2 2 , y0! 5-38 Transformations of Random Variables (cont.) • The reason the “obvious” procedure failed is that the PDF has no meaning outside of an integral! • The correct procedure is to transform the CDF and then compute its derivative to get the transformed PDF. • Let Y = g(X) X = g-1(Y) be a one-to-one “mapping”, then 1 d g ( y) fY ( y ) f X ( g 1 ( y )) dy • For X ~ exponential and Y X X = Y 2 then y2 2 2y 2 d y fY ( y ) f X ( y ) e , y0 dy • The scaling factor looks different from the Rayleigh PDF. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-39 Transformations of Random Variables (cont.) • Why is a one-to-one mapping is important? Let X ~ N(0, x2) and apply the transformation Y = X 2 (Y should be chi-square) y d y 1 2 x2 fY ( y ) f X ( y ) e , y0 2 dy 2 2 x y • The above “PDF” does not integrate to 1! Instead, it integrates to ½. 0.4 4 0.3 3 f (y) 0.2 Y X f (x) • What went wrong? Two points from X map into Y 0.1 0 2 1 -4 EMIS7300 Fall 2005 -3 -2 -1 0 x 1 2 3 4 0 Copyright 2003-2005 Dr. John Lipp 1 2 3 4 5 y 6 7 8 9 5-40 Transformations of Random Variables (cont.) • In general, a mapping of X to Y with a function Y = g(X) must be analyzed by dividing g(X) into N monotonic regions (roots) and then summing the PDF contributions from each region N d g 1 ( y ) 1 fY ( y ) f X ( g ( y )) dy i 1 • The transformation Y = X 2 has two monotonic regions, X < 0 and X 0 (the equality belongs on the right). 25 20 x2 15 10 5 0 EMIS7300 Fall 2005 -4 -3 -2 -1 0 x 1 Copyright 2003-2005 Dr. John Lipp 2 3 4 5-41 Transformations of Bivariate Random Variables • The process is identical to that for a random variable except that the derivative operation is replaced with the Jacobian. • Let Y1 = g1(X1, X2) and Y2 = g2(X1, X2). fY1Y2(y1,y2) is found with N fY1 ,Y2 ( y1 , y2 ) i 1 The joint PDF x1 , x2 f X1 , X 2 ( g ( y1 , y2 ), g ( y1 , y2 )) J y1 , y2 1 1 1 2 where g11 ( y1 , y2 ) x1 , x2 y1 J 1 g y , y 2 ( y1 , y 2 ) 1 2 y1 EMIS7300 Fall 2005 g11 ( y1 , y2 ) g1 ( x1 , x2 ) y2 x1 1 g 2 ( y1 , y2 ) g 2 ( x1 , x2 ) x1 y2 Copyright 2003-2005 Dr. John Lipp g1 ( y1 , y2 ) x2 g 2 ( x1 , y2 ) x2 5-42 1 Transformations of Bivariate Random Variables (cont.) • Example: Let X1 and X2 be zero-mean, independent Gaussian random variables with equal variances. Compute the PDF fR,(r,) of the polar transform R X 12 X 22 , tan 1 ( X 1 X 2 ). • First, note that this transform is one-to-one. • Second, the PDF of fX1,X2(x1,x2) is 1 2 x12 x22 1 2 x f X1 , X 2 ( x1 , x2 ) e 2 x2 • Third, the Jacobian is x x x1 , x2 x1 J 1 y1 , y2 tan ( x1 x2 ) x1 2 1 EMIS7300 Fall 2005 2 2 x x x2 tan 1 ( x1 x2 ) x2 2 1 2 2 Copyright 2003-2005 Dr. John Lipp 1 x1 r x2 r2 x2 r x1 2 r 1 r 5-43 Transformations of Bivariate Random Variables (cont.) • Substituting x1 , x2 1 1 fY1 ,Y2 ( y1 , y2 ) f X1 , X 2 ( g1 (r , ), g 2 (r , )) J r , f X1 , X 2 (r cos( ), r sin( )) r 1 2 r 2 cos2 ( ) r 2 sin 2 ( ) 1 2 x e r 2 2 x r 2 1 r 2 e 2 x 2 x • Thus R ~ Rayleigh with = x2 and ~ uniform [0,2]. • Moreover, R and are statistically independent. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-44 Transformations of Bivariate Random Variables (cont.) • The process of transformation of random variables has several important and useful results. • A random variable U ~ uniform [0,1] can be transformed to any other PDF fX(x) with the transform X = FX-1(U). – Exponential: X = - ln(1 - U). – Rayleigh: X ln(1 U ). The only limitation is being able to invert FX(x). • A pair of independent, zero-mean, unit variance Gaussian random variables can be generated from X1 = Rcos() and X2 = Rsin() where R is Rayleigh ( = 1) and is uniform [0,2]. EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-45 Transformations of Bivariate Random Variables (cont.) • Let X1 and X2 be independent random variables and define Y = X1 + X2 W = X1 – The transformation is one-to-one. – The Jacobian is x x x x 1 1 2 1 2 1 1 1 x ,x x1 x2 J 1 2 1 x1 x1 1 0 y, w x1 x2 – Thus fY,W(y,w) = fX1 (w)f X2(y-w). – Integrating vs. w fY ( y ) f X1 ( w) f X 2 ( y w)dw f X1 ( x1 ) f X 2 ( x2 ) EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-46 Transformations of Bivariate Random Variables (cont.) • Let X1 and X2 be random variables and define Y = X1 X2 W = X2 – The transformation is one-to-one. – The Jacobian is x x x x 1 x ,x J 1 2 y, w – Thus and EMIS7300 Fall 2005 1 2 x1 x2 x1 1 2 x2 x2 x2 x2 x1 0 1 1 1 1 x2 w fY,W(y,w) = fX1,X2(y / w, w) / |w| 1 fY ( y ) f X1 , X 2 ( y / w, w) dw w Copyright 2003-2005 Dr. John Lipp 5-47 Transformations of Bivariate Random Variables (cont.) • Let X1 and X2 be random variables and define Y = X1 / X2 W = X2 – The transformation is one-to-one. – The Jacobian is 1 x1 x2 x1 x2 1 x ,x x1 x2 J 1 2 x2 x2 x2 y, w 0 x1 x2 – Thus x1 x22 1 1 x2 w fY,W(y,w) = fX1,X2(yw,w) and fY ( y ) f X1 , X 2 ( yw, w) w dw EMIS7300 Fall 2005 Copyright 2003-2005 Dr. John Lipp 5-48 Homework • Mandatory (answers in the back of the book): 5-27 EMIS7300 Fall 2005 5-37 5-39 5-89 Copyright 2003-2005 Dr. John Lipp 5-49