Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

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1 Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the eal line. Suppose g maps A onto B, so that thee is an inese map x = hy) fom B back to A. Kenneth Hais Depatment of Mathematics Uniesity of Michigan Apil 3, 009 Let X be a continuous andom aiable with known density f X x). Let Y = GX). Then the density of Y is ) f Y y) = f X hy) d. dt hy) Note: Compae to Ross, Theoem 5.7., page 43. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Poblem Two Functions of Two Random Vaiables Two Functions of Two Random Vaiables Definition Nice Tansfomations Let the continuous andom aiables X, Y ) hae joint density f X,Y x, y) and let A = x, y) : f X,Y x, y) > 0}. X, Y ) detemines a point with xy-coodinates in the egion A. Conside the continuous andom aiables U, V ) gien by U = g X, Y ) V = g X, Y ). U, V ) detemines a point with u-coodinates in some egion B. Poblem. If the tansfomation fom xy-coodinates to u-coodinates gien by u = g x, y) = g x, y). is nice on A, then we can poduce the joint density f U,V u, ) fo the andom aiable U, V ). Definition A tansfomation fom xy-coodinates to u-coodinates xy u) gien by is nice on A, if u = g x, y) The patial deiaties u on A. x, u y, x = g x, y)., and y The Jacobian of the tansfomation is nonzeo on A: u u x y Jx, y) = = u x y u y x 0 whenee x, y) A. x y exist and ae continuous Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

2 Two Functions of Two Random Vaiables Change of coodinates A nice tansfomation on A xy u) amounts to simply a change of coodinates of the plane fom xy-coodinates to u-coodinates. We can ecoe the oiginal xy-coodinates fom the new u-coodinates. Suppose xy u) is nice tansfomation on A u = g x, y) = g x, y) to u-coodinates on a egion B. Thee is a eese tansfomation u xy) fom u-coodinates to xy-coodinates x = h u, ) y = h u, ). which maps B onto A and which ae also nice on B. Jacobians Two Functions of Two Random Vaiables The Jacobian of the oiginal tansfomation xy u) is the deteminant u u x y Jx, y) = = u x y u y x x The Jacobian of the inese tansfomation u xy) is the deteminant x x Ju, ) = u y y = x y u x y u Since xy u) is nice on A, Jx, y) 0 whenee x, y) A and Ju, ) 0 whenee u, ) B. u Futhemoe, the two Jacobian deteminants ae ineses y Jx, y) = Ju, ) Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Main Theoem Theoem Two Functions of Two Random Vaiables Let X, Y ) be continuous andom aiables with joint density f X,Y x, y), and U, V ) be andom aiables gien by U = g X, Y ) V = g X, Y ). Suppose the xy u) tansfomation u = g x, y) is nice on A = x, y) : f X,Y x, y) 0}. Let the inese u xy) fom B to A be x = h u, ) = g x, y). y = h u, ). Two Functions of Two Random Vaiables Pictue of Theoem f U,V u, ) du d P U, V ) B} UV P U,V B B u, P X, Y ) A} f X,Y x, y) Ju, ) du d d XY P X,Y A A x,y The joint density of U, V ) is gien fo u, ) B by eithe equation f U,V u, ) = f X,Y h u, ), h u, ) ) Ju, ) f U,V u, ) = f X,Y h u, ), h u, ) ) Jx, y) whichee is moe conenient to compute. du Aea of B du d Aea of A J u, du d Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

3 Two Functions of Two Random Vaiables Sketch of Poof of Theoem Let B B and suppose u xy) maps B to A A. P U, V ) B} = P X, Y ) A} = f X,Y x, y) dx dy = x,y) A u,) B f X,Y h u, ), h u, ) ) Ju, ) du d using the Change of Vaiables Theoem of analysis. Intuitiely, we can beak B into small egions B which u xy) tansfoms to small egions A of A whee fo any u, ) B: f U,V u, ) Aea B ) f X,Y h u, ), h u, )) Aea A ) whee Aea A ) Aea B ) Ju, ). Diffeentiate the integals to get the tansfomation ule: f X,Y h u, ), h u, ) ) Ju, ) if u, ) B f U,V u, ) = 0 othewise. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3 Functions of a Random Vaiable: Density. Let X and Y be continuous andom aiables with joint density f X,Y x, y) and whee X 0. Conside U = XY V = X. The tansfomation xy u) is gien by u = xy = x. The inese tansfomation u xy) is gien by x = y = u. The Jacobian fo tansfomation fo u xy) is Ju, ) = 0 u = Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Functions of a Random Vaiable: Density Rectangle to Pola coodinates So, the joint density is f U,V u, ) = f X,Y h u, ), h u, ) ) Ju, ) u ) = f X,Y, We can compute the maginal f U u) = f XY u) by f XY u) = u ) f X,Y, d It is often conenient to change fom ectangula coodinates xy to pola coodinates θ. The tansfomation xy θ) is whee > 0 and π < θ π. = x + y θ = actan y x. The inese tansfomation θ xy) fom pola back to ectangula is x = cos θ y = sin θ. The tansfomation is xy θ) nice in the punctued plane R 0, 0)}. Veified in thee slides. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

4 Coneting Rectangle to Pola Coneting Rectangle to Pola Rectangle xy-coodinates to pola θ-coodinates: Plot of tan y x on [ π, π]. The fou quadants of the plane ae = x + y, > 0 θ = actan y, π < θ π, x Pola θ-coodinates to ectangle xy-coodinates x = cos θ y = sin θ < x, y <. x,y,θ I : x, y > 0 II : x < 0, y > 0 III : x, y < 0 IV : x > 0, y < 0 II III 6 I IV 4 x y y sinθ Π Π Π Π 4 Θ actan y x 6 x cosθ Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Poblem: Rectangle to Pola Poblem. Let X, Y ) be andomly chosen in some egion R of the xy-plane with joint density f X,Y x, y). Conside the andom aiables giing the pola coodinates R = X + Y whee R > 0 and π < Θ π. Θ = actan Y X The Jacobian is easiest to compute on the θ-plane: J, θ) = cos θ sin θ sin θ cos θ = cos θ + sin θ = The joint distibution of R, Θ is f R,Θ, θ) = f X,Y cos θ, sin θ) > 0, π < θ π. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3. Let X, Y ) be unifomly distibuted in R = unit cicle. So, So, f X,Y x, y) = π f R,Θ, θ) = f X,Y cos θ, sin θ) = π The maginals ae f R ) = f Θ θ) = π π 0 when x + y. dθ = 0 <, π π d = π Thus, Θ is unifomly distibuted on π, π]. 0 <, π < θ π π < θ π. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

5 . Let X, Y ) be independent and nomally distibuted in the plane with µ = 0, σ ). So, f X,Y x, y) = πσ e x +y )/σ Since f R,Θ, θ) = f X,Y cos θ, sin θ), f R,Θ, θ) = πσ e cos θ+ sin θ)/σ = πσ e /σ 0 <, π < θ π. The maginals ae f R ) = f Θ θ) = π π πσ e 0 πσ e /σ dθ = σ e /σ 0 <, /σ d = π π < θ π. Thus, Θ is unifomly distibuted on π, π] and R is the Rayleigh distibution the distance of X, Y ) fom the oigin). Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3. Let R be exponentially distibuted with mean and Θ be unifomly distibuted in π, π], both independent. The joint distibution is f R,Θ, θ) = π e / 0 <, π < θ π Let X and Y be andom aiables detemined by X = R cos Θ Y = R sin Θ Sole fo, θ in the tansfomation x = cos θ and y = sin θ: = x + y θ = actan y x. The Jacobian deteminant is easiest to compute using θ-coodinates: cos θ J, θ) = sin θ = cos θ + sin θ = sin θ cos θ. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, 009 / 3 So, f R,Θ, θ) = π e / 0 <, π < θ π f X,Y x, y) = f R,θ x + y, actan y x ) = π e x +y )/ X and Y ae independent and nomally distibuted andom aiables with µ = 0, σ = ). The maginals ae obtained by integating f X,Y x, y): f X x) = f Y y) = π e x / π e y / continued Let U and V be unifomly distibuted on 0, ). Conside the andom aiable Θ: Θ = πv π So, Θ is unifomly distibuted on π, π). Conside the andom aiable R: R = ln U By Poposition 5.7. Ross, page 43), soling, u = e /. f R ) = f U e / ) u = e / So, R is exponentially distibuted with mean. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

6 Coneting Rectangle to Pola Let X and Y be andom aiables detemined by X = R cos Θ Y = R sin Θ Simulating a standad nomal andom aiable with a pai of independent unifom andom aiables on 0, ). Then X and Y ae independent standad nomal andom aiables!! We can simulate a standad nomal andom aiable X by using two independent unifom andom aiables U and V on 0, ): X = ln U cos πv π ). 000 data points ,000 data points Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 of Joint Distibution. Let X and Y be independent and unifomly distibuted on 0, ]. Find the joint pobability density function fo the andom aiables U = X Y V = XY. Indiidually, the distibution of X and Y ae if 0 x if 0 y f X x) = f Y y) = 0 othewise 0 othewise So, the joint distibution f X,Y x, y) is f X,Y x, y) = f X x) f Y y) = if 0 x, y 0 othewise. of Joint Distibution The tansfomation into u-coodinates u = x y is one-to-one and has an inese = xy, x = u y = u. The Jacobian deteminant is easiest when computed in xy coodinates: Jx, y) = y x y y x = x y = u So, u, > 0 and ) f U,V u, ) = f X,Y u, = u if 0 < u, u u u 0 othewise. Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

7 of Joint Distibution of Joint Distibution It emains to compute the bounds on u and. 0 < u, u = 0 < u, 0 < u. Only one of these anges need be etained, depending upon whethe u 0, ] o u [, ): u if 0 < u <, 0 < u, f U,V u, ) = o, if u, 0 < u, 0 othewise. u if 0 < u <, 0 < u, f U,V u, ) = o, if u, 0 < u, 0 othewise. We compute the maginals. f U u) = f ) = u 0 u d if 0 < u < d if 0 < u < u 0 u d if u = d if u u 0 othewise 0 othewise u du if 0 < 0 othewise = ln if 0 < 0 othewise Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3 of Joint Distibution Plot of aea detemined by 0 < u < = 0 < u and u = 0 < u u Kenneth Hais Math 45) Math 45 Into to Pobability Lectue 30 Apil 3, / 3

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