Lecture 8: Jointly distributed random variables

Transcription

1 Lecture : Jointly distributed random variables

2 Random Vectors and Joint Probability Distributions Definition: Random Vector. An n-dimensional random vector, denoted as Z = (Z, Z,, Z n ), is a function from the sample space Ω to R n, the n-dimensional real space. For each outcome ω Ω, Z(ω) is a n-dimensional real-valued vector and is called a realization of the random vector Z. Example: Bivariate Random Vector. Consider a bivariate r.v. Z = (X, Y ). For each outcome ω Ω, Z(ω) is a pair (x, y) R.

3 Discrete Bivariate Random Vector Example : toss two fair dice. X = sum of two dice Y = difference of the two dice Probability of events in terms of (X, Y ). X Y

4 Discrete Bivariate Random Vector Definition Joint probability mass function of (X, Y ): Let (X, Y ) be discrete bivariate random vector. Then the function f (x, y) from R to R defined by f (x, y) = P (X = x, Y = y) is called the joint probability mass function of (X, Y ). Remark: P (X = x, Y = y) = P (X = x and Y = y) Remark: f (x, y) is written as f X,Y (x, y) to identify that it is the pmf of (X, Y ). Remark: f (x, y) completely defines the probability distribution of the random vector (X, Y ). P ((X, Y ) A) = f (x, y) Properties of f X,Y (x, y) : () f X,Y (x, y) for all (x, y) R, () (x,y) R f X,Y (x, y) =. (x,y) A

5 Expectation of Function of Discrete Bivariate Random Vector g (x, y) be a real function defined for all possible values (x, y) of (X, Y ). E [g (x, y)] = Continuation of Example : g (x, y) = xy E [g (X, Y )] = E [XY ] =? (x,y) R g (x, y) f (x, y)

6 Marginal Distribution of Discrete Bivariate Random Vector Theorem Let (X, Y ) be a discrete bivariate random vector with joint pmf f X,Y (x, y). Then the marginal pmfs of X and Y, f X (x) = P (X = x), f Y (y) = P (Y = y), are given by f X (x) = y R f X,Y (x, y) f Y (y) = x R f X,Y (x, y)

7 Marginal Distributions of Discrete Bivariate Random Vector Example: Tossing a coin three times. Let X = number of heads in all three tosses and Y = number of tails in the last two tosses. The joint distribution and marginal distributions of (X, Y ) are as follows (X \ Y ) X 3 Y 3 3 4

8 Marginal Distributions of Discrete Bivariate Random Vector Remark: For bivariate random variable (X, Y ), the joint distribution contains more information than the marginal distributions of X and Y separately. Given a joint distribution, we are able to recover the marginal distributions of X and of Y, but not conversely. Example: Toss a coin twice. Let X = number of heads in the first toss, Y = number of tails in the first toss, and Y = number of tails in the second toss. The marginal distributions of (X, Y = Y ) and (X, Y = Y ) are the same, but their joint distributions are different.

9 Joint Cumulative Distribution Function Definition: Joint Cumulative Distribution Function. The function of two real variables, defined as F X,Y (x, y) = P{X x, Y y} is called the joint cumulative distribution function of bivariate random variable (X, Y ). Properties of joint CDF F X,Y (x, y): limx F X,Y (x, y) = lim y F X,Y (x, y) = and lim x,y F X,Y (x, y) =. For each given y, the function F X,Y (x, y) is nondecreasing and continuous from the right in x. For each given x, the function FX,Y (x, y) is nondecreasing and continuous from the right in y. For all x x and y y, F XY (x, y ) F XY (x, y ) F XY (x, y ) + F XY (x, y ). Why?

10 Joint Cumulative Distribution Function Definition: Joint Cumulative Distribution Function. The function of two real variables, defined as F X,Y (x, y) = P{X x, Y y} is called the joint cumulative distribution function of bivariate random variable (X, Y ). Properties of joint CDF F X,Y (x, y): limx F X,Y (x, y) = lim y F X,Y (x, y) = and lim x,y F X,Y (x, y) =. For each given y, the function F X,Y (x, y) is nondecreasing and continuous from the right in x. For each given x, the function FX,Y (x, y) is nondecreasing and continuous from the right in y. For all x x and y y, F XY (x, y ) F XY (x, y ) F XY (x, y ) + F XY (x, y ). Why? Because P(x < X x, y < Y y ) = F X,Y (x, y ) F X,Y (x, y ) F X,Y (x, y ) + F X,Y (x, y ).

11 Continuous Bivariate Random Vector and Continuous Joint Distribution Definition: Random variables (X, Y ) are jointly continuous (or have joint continuous distribution) if there exists a function f XY (x, y) such that for every a, b, we have F XY (a, b) = P{X a, Y b} = b a The function f X,Y (x, y) is called joint (or bivariate) probability density function of (X, Y ). f XY (x, y) dxdy.

12 Continuous Bivariate Random Vector and Continuous Joint Distribution Remarks: Function f (x, y) is a pdf of a bivariate random variable if and only if f (x, y) and f (x, y) dxdy =. The definition implies for any Borel set B R on the plane, P{(X, Y ) B} = f X,Y (x, y) dxdy. B Geometric interpretation of P{(X, Y ) B}.

13 Continuous Bivariate Random Vector and Continuous Joint Distribution Remarks: Suppose f X,Y (x, y) is continuous on (x, y), f X,Y (x, y) P{(X, Y ) (x.5ε, x+.5ε) (y.5δ, y+.5δ)} εδ when ε, δ are small.

14 Continuous Bivariate Random Vector and Continuous Joint Distribution Remarks: For continuous r.v. (X, Y ), pdf fx,y (x, y) can be recovered from cdf F X,Y (x, y) by f X,Y (x, y) = x y F X,Y (x, y). We can generalize the bivariate concepts in the continuous case to the multivariate concepts: The joint cdf in the continuous case is given by F (x,, x n ) = xn x f (u,, u n )du du n. Also, the joint pdf can be recovered from the joint cdf by f (x,, x n ) = n x x n F (x,, x n ).

15 Continuous Bivariate Random Vector and Continuous Joint Distribution Example: The density f (x, y) is given by the formula { c x(x + y) if x, y, x + y <, f (x, y) = otherwise. Find (a) P(X /); (b) P(X = 3Y ).

16 Continuous Bivariate Random Vector and Continuous Joint Distribution ANS: (a) Because x so c =. Then f (x, y) dxdy = P(X /) = (b) P(X = 3Y ) =. = c = c = c / x x c x(x + y) dydx x ( x) + x( x) dx x( x)(x + ( x)) dx x x 3 dx = c, x(x + y) dydx = 7/6.

17 Continuous Bivariate Random Vector and Continuous Joint Distribution Example: Suppose f XY (x, y) = cx y for x y. Find (a) the value of c; (b) P(X > Y ). ANS: (a) f X,Y (x, y) > for [, ] [x, ], = x cx y dydx = = c c x y x dx (x x 6 )dy = c( 3 7 ). (b) Thus c = 4. P(X > Y ) = x x cx y dydx = c ( 5 7 ) = 3.

18 Continuous Bivariate Random Vector and Continuous Joint Distribution Example: Suppose (X, Y ) has a joint cdf F XY (x, y) = 6xy(x + y) for x and y. Find (a) f X,Y (x, y); (b) P( X, Y ). ANS: (a) For x and y, (b) We have f X,Y (x, y) = F X,Y (x, y) x y = (x + y). or P( X, Y ) = (x + y) dxdy = 3, P( X, Y ) = F X,Y (, ) F XY (, ) F XY (, ) +F X,Y (, ) = 3.

19 Marginal Distributions of Continuous Bivariate Random Vector Definition: Continuous Marginal Distribution. If (X, Y ) is a pair of continuous random variables with bivariate probability density function f X,Y (x, y), then the functions f X (x) = f Y (y) = f X,Y (x, y)dy f X,Y (x, y)dx are called the marginal pdf of variables X and Y, respectively.

20 Marginal Distributions of Continuous Bivariate Random Vector Example: A man shoots at a circular target. Assume that he is certain to hit the target. However, he is unable to aim with any more precision, so that the probability of hitting a particular part of the target is proportional to the area of this part. Let X and Y be the horizontal and vertical distances from the center of the target to the point of impact. ANS: The joint distribution of (X, Y ) is { f X,Y (x, y) = πr x + y R, otherwise.

21 Marginal Distributions of Continuous Bivariate Random Vector The marginal distribution of X is f X (x) = = = R x f (x, y)dy f (x, y)dy R x R x R x πr dy = R x πr.

22 Marginal Distributions of Continuous Bivariate Random Vector Example: Suppose f X,Y (x, y) = cy for x y. Find (a) f X (x); (b) f Y (y). ANS: For x, f X (x) = x cy dy = c 3 ( x 6 ). Because f X (x)dx =, so c = 7 4, f X (x) = 7 ( x 6 ). For y, f Y (y) = y y cy dx = 7 y 5/.

23 Marginal Distributions of Continuous Bivariate Random Vector Example: Suppose (X, Y ) follows bivariate Normal distribution N(µ, µ ; σ, σ, ρ), i.e., = then f X,Y (x, y) πσ σ ρ { [ (x ) ( ) ( ) µ x µ y µ exp ( ρ ρ + ) σ σ σ ( ) ]} y µ, σ f X (x) = f Y (y) = πσ exp πσ exp { (x µ ) σ { (y µ ) σ }, }.

24 Marginal Distributions of n-dimensional random vector Multivariate Marginal Distribution: When there are more than two random variables, we can define not only the marginal distributions of individual random variables but also the joint marginal distributions of any subset of random variables. Let X = (X, X,, X n ) be a n-dimensional random vector. Divide the random variables in X into Y and Z. In the discrete case, the marginal distribution of Y is given by f Y (y) = P(Y = y) = z P(Y = y, Z = z). In the continuous case, the marginal density of Y is given by f Y (y) = z f YZ (y, z) dz.

25 Independence Definition: Independence. We say that random variables X and Y are independent if events {X A} and {Y B} are independent for all Borel sets A, B, that is P(X A, Y B) = P(X A)P(Y B). Theorem: The random variables X and Y are independent if and only if their joint and marginal cdf s satisfy the following condition: For every x, y R, we have F X,Y (x, y) = F X (x)f Y (y), where F X (x) and F Y (y) are marginal cdf s of X and Y, respectively.

26 Independence Remarks: For discrete bivariate random variable (X, Y ), independence is implied by P(X = x, Y = y) = P(X = x)p(y = y), for every (x, y). For continuous bivariate random variable (X, Y ), suppose the joint pdf f XY (x, y) is continuous on its support. Then independence is implied by f X,Y (x, y) = f X (x)f Y (y), where f X,Y (x, y) is the joint density, f X (x) and f Y (y) are the marginal densities of X and Y, respectively.

27 Independence Theorem: The two random variables X and Y are independent if and only if the joint pmf/pdf can be written as f X,Y (x, y) = h(x)g(y), for all < x, y <. Remark: h(x) and g(y) are not necessary pdf/pmf.

28 Independence Example: Let the experiment consist of three tosses of a coin, and let X = number of heads in all three tosses and Y = number of tails in the last two tosses. The joint and marginal distributions of (X, Y ) are (X, Y ) X 3 Y P(X = 3, Y = ) = P(X = 3)P(Y = ). X and Y are not independent.

29 Independence Example: Consider the case of three tosses of a coin. Let U be the number of heads in the first two tosses, and let V be the number of tails in the last toss. The sample points are S U V S U V HHH HTT HHT THT HTH TTH THH TTT

30 Independence The joint and marginal probability distributions are (U, V ) U V U and V are independent. 4 4

31 Independence Example: The joint density of random variables X and Y is { cx n e αx βy x >, y >, f (x, y) = otherwise. Are X and Y independent? Example: The joint density of random variables X and Y is { cx n e αx βy x >, y >, x + y <, f (x, y) = otherwise. Are X and Y independent?

32 Sums of Independent Random Variables Question: Suppose that X and Y are independent continuous random variables having the pdf f X (x) and f Y (y). Find the distribution of Z = X + Y. The CDF of Z is obtained as follows: F Z (a) = P(X + Y a) = f X (x)f Y (y)dxdy = = = x+y a a y f X (x)f Y (y)dxdy f X (x)dxf Y (y)dy F X (a y)f Y (y)dy

33 Sums of Independent Random Variables By differentiating the CDF F X +Y (a) with respect to a, we obtain the pdf f X +Y (a) is given by f Z (a) = = d da = = F X (a y)f Y (y)dy d da F X (a y)f Y (y)dy f X (a y)f Y (y)dy

34 Sums of Independent Random Variables Sums of two independent uniform random variables Let the joint density of (X, Y ) be { < x <, < y <, f X,Y (x, y) =. otherwise. Find the pdf of Z = X + Y. ANS: The marginal pdf of X and Y are { < a < f X (a) = f Y (a) = otherwise For a, f Z (a) = For < a <, f Z (a) = Otherwise, fz (a) =. f X (a y)f Y (y) dy = f X (a y)f Y (y) dy = a a dy = a. dy = a.

35 Sums of Independent Random Variables Sums of n independent uniform random variables suppose that X, X,..., X n are independent uniform (, ) random variables, F n (x) = P{X + + X n x}, x ANS: F n (x) = x n n!, x By mathematical induction, F n (x) = F n (x) = x n (n )!, x n n X i = X i + X n i= i= = (n )! = x n n! F n (x y)f Xn (y)dy x (x y) n dy

36 Sums of Independent Random Variables Sum of Gamma Random Variables. If X and Y are independent Gamma random variables with respective parameters (s, λ) and (t, λ), then X + Y is a Gamma random variable with parameters (s + t, λ). Proof f (y) = λe λy (λy) t, y Γ(t) f X +Y (a) = = = = a λe λ(a y) [λ(a y)] s λe λy (λy) t dy Γ(s)Γ(t) λ s+t a Γ(s)Γ(t) e λa (a y) s y t dy by letting x = y a λ s+t Γ(s)Γ(t) e λa a s+t ( x) s x t dx λ s+t Γ(s)Γ(t) e λa s+t Γ(s)Γ(t) a Γ(s + t)

37 Sums of Independent Random Variables Extension: If X i, i =,..., n are independent gamma random variables with respective parameters (t i, λ), i =,, n, then n i= X i is gamma with parameters ( n i= t i, λ). Sum of Exponential Random Variables. Let the joint density of (X, Y ) be { e (x+y)/β x, y, f XY (x, y) = β. otherwise. Find the pdf of Z = X + Y. ANS: exp(β) Gamma(,β),Z=X+Y Gamma(,β)

38 Sums of Independent Random Variables If Z, Z,, Z n are the independent standard normal distribution, n Y = Zi, i= then Y follows Chi-squared distribution with m degrees of freedom, denoted by Y χ n. f Z (y) = = = y [f z( y) + f z ( y)] e y/ y π e y/ (y/) / π Z Gamma(, ) Y Gamma( n, )

39 Sums of Independent Random Variables Sums of Independent Normal Random Variables If X i, i =,..., n are independent normal random variables with respective parameters (µ i, σi ), i =,, n, then n i= X i is normal with parameters ( n i= µ i, n i= σ i ). Sums of independent Poisson Random Variables If X and Y are independent Poisson random variables with respective parameters λ and λ, the distribution of X + Y is Poisson random variables with parameters λ + λ. Sums of independent Binomial Random Variables Let X and Y be independent binomial random variables with respective parameters (n, p) and (m, p), the distribution of X + Y is Binomial(m + n, p).

40 Order Statistics Let X, X,..., X n be n independent and identically distributed continuous random variables having a common density f and distribution function F. Define X () = smallest of X, X,..., X n X () = second smallest of X, X,..., X n X (j) = jth smallest of X, X,..., X n X (n) = largest of X, X,..., X n The ordered values X () X () X (n) are known as the order statistics corresponding to the random variables X, X,, X n. In other words, X (),, X (n) are the ordered values of X,, X n.

41 Order Statistics f X(j) (x) = n! (n j)!(j )! [F (x)]j [ F (x)] n j f (x) f X() (x) = n[ F (x)] n f (x) f X(n) (x) = n[f (x)] n f (x)

42 Conditional Distributions Definition: Discrete Conditional Distribution. Let (X, Y ) be a discrete bivariate random variable. For y such that the marginal probability P(Y = y) >, f X Y (x y) = P(X = x Y = y) = P(X = x, Y = y) P(Y = y) = f X,Y (x, y). f Y (y) Similarly, for x such that the marginal probability P(X = x) >, f Y X (y x) = P(Y = y X = x) = Remarks: P(X = x, Y = y) P(X = x) = f X,Y (x, y). f X (x) For y such that f Y (y) =, conditional pmf f X Y (x y) is not defined. For y such that fy (y) >, we have f X Y (x y) and x f X Y (x y) =. f X Y (x y) can be considered as a pmf.

43 Conditional Distributions Example: Suppose two random variables X and Y are independent, and X Poisson(λ ), Y Poisson (λ ). We can verify that X + Y P(λ + λ ). Find the conditional distribution of X given X + Y = n. Sol. By the definition of conditional distributions, for k =,,,, n, we have P(X = k X + Y = n) = = = = P(X = k, X + Y = n) P(X + Y = n) P(X = k)p(y = n k) P(X + Y = n) λ k k! e λ λn k (n k)! e λ (λ +λ ) n e n! (λ +λ ) n! λ k λn k k!(n k)! (λ + λ ) n ( ) λ k ( λ ) n k = C k n λ + λ λ + λ ( ) = C k λ k ( n λ ) n k λ + λ λ + λ Therefore, given X + Y = n, X follows a binomial distribution B(n, p) with p = λ λ +λ.

44 Conditional Distributions Example: Let the experiment consist of two tosses of a die. We let X and X denote the result of the first and the second toss, respectively, and put Z = X X. The joint distribution of X and Z are as follows (X, Z) X Z P(X odd Z = 5) = /, P(Z > X = 5) = /

45 Conditional Distributions Definition: Conditional Probability Density Function. For continuous bivariate random variable (X, Y ), the conditional probability density functions f X Y and f Y X are defined by provided that f Y (y) >, and f X Y (x y) = f X,Y (x, y) f Y (y) f Y X (y x) = f X,Y (x, y) f X (x) provided that f X (x) >, where f X,Y (x, y) is the joint density of (X, Y ) and f X and f Y are the marginal densities of X and Y, respectively. Remark: For given y such that f Y (y) >, f X Y (x y) can be considered as a pdf; similarly, for given x such that f X (x) >, f Y X (y x) can be considered as a pdf.

46 Conditional Distributions Remarks: Assume for simplicity that fx,y (x, y), and hence also f X (x), f Y (y), are continuous. Then for any Borel sets A, B R, lim P(X A y Y y + h) h = y+h f A y X,Y (u, v) dv du y+h f y Y (v)dv = A f X,Y (u, y) du f Y (y) = f X Y (u y)du, A lim P(Y B x X x + h) = h = = x+h B x f X,Y (u, v) du dv x+h f x X (u)du B f X,Y (x, v) dv f X (x) f Y X (v x)dv. B

47 Conditional Distributions Example: Let the joint density of (X, Y ) be given by { cxy x, y, x + y, f X,Y (x, y) = otherwise. Find f X Y (x y) and f Y X (y x). ANS: For x, f X (x) = x and for y, f Y (y) = y cxy dy = c 3 [x x( x)3 ] = c 3 (3x 3x 3 + x 4 ), cxy dx = c [y y ( y) ] = c (y 3 y 4 ).

48 Conditional Distributions For given < y <, f X Y (x y) = f X,Y (x, y)/f Y (y) = x y y for y x ; f X Y (x y) = otherwise. For given < x <, f Y X (y x) = f XY (x, y)/f X (x) = 3y 3x 3x + x 3 for x y ; f Y X (y x) = otherwise.

49 Conditional Distributions Example: A point X is chosen at random from the interval [A, B] according to the uniform distribution, and then a point Y is chosen at random, again with uniform distribution, from the interval [X, B]. Find the marginal distribution of Y. ANS: For A x y B, For A y B, f X,Y (x, y) = f X (x)f Y X (y x) = (B A)(B x). f Y (y) = = y A (B A)(B x) dx ( ) B A B A log. B y

50 Conditional Distributions Example: Assume that Y takes one of n integer values with P(Y = i) = p i and p + + p n =. For a given Y = i, the random variable X has a continuous distribution with density φ i (x). Find the marginal distribution of X. ANS: P(X x) = = n P(X x Y = i)p(y = i) i= n x p i i= φ i (u)du. The pdf of X is f X (x) = n i= p iφ i (x).

51 Conditional Distributions Example: Suppose (X, Y ) are bivariate normally distributed, then The conditional distribution of X given Y = y is a normal distribution, i.e. X Y = y N(µ + ρσ σ (x µ ), σ ( ρ )) Similarly, the conditional distribution of Y given X = x is as follows: Y X = x N(µ + ρσ σ (x µ ), σ ( ρ ))

52 Joint Probability Distributions of Functions of Random Variables Let X and X be jointly continuous random variables with joint probability density function f X,X. It is sometimes necessary to obtain the joint distribution of the random variables Y and Y, which arise as functions of X and X. Specifically, suppose that Y =g (X, X ) and Y = g (X, X ) for some functions g and g. Assume that the functions g and g satisfy the following conditions: For (x, x ) A, the equations y = g (x, x ) and y = g (x, x ) can be uniquely solved for x and x in terms of y and y, with solutions given by, say, x = h (y, y ), x = h (y, y ), (y, y ) B.(g, g defines a one-to-one transformation of A onto B)

53 Joint Probability Distributions of Functions of Random Variables The functions g and g have continuous partial derivatives at all points (x, x ) and are such that the determinant g g J(x, x ) = x x = g g g g x x x x g x g x f Y Y (y, y ) = f X,X (x, x ) J(x, x )

54 Joint Probability Distributions of Functions of Random Variables Example Let X and X be jointly continuous random variables with probability density function f X,X. Let Y = X + X, Y = X X. Find the joint density function of Y and Y in terms of f X,X. ANS Let g (x, x ) = x + x and g (x, x ) = x x. Then J(x, x ) = = x = y + y, x = y y f Y,Y = f X,X ( y + y, y y )

55 Joint Probability Distributions of Functions of Random Variables Example Suppose X N(µ, σ ), Y N(µ, σ ), and X, Y are independent. Put U = X + Y, V = X Y. Find the joint pdf of (U, V ) and show that U, V are independent. ANS: From U = X + Y, V = X Y, we have X = (U + V ), Y = (U V ).

56 Joint Probability Distributions of Functions of Random Variables It follows that f U,V (u, v) = f X,Y ( (u + v), (u v)) = πσ e σ [(x µ) +(y µ) ] = = = 4πσ e σ (u+v µ) e σ (u v µ) 4πσ e 4σ (u µ) e 4σ v πσ e 4σ (u µ) πσ e 4σ v for < u, v It follows that U N(µ, σ ), V N(, σ ) and U and V are independent.

57 Joint Probability Distributions of Functions of Random Variables Example If X and Y are independent gamma random variables with parameters (α, λ) and (β, λ), respectively, compute the joint density of U = X + Y and V = X /(X + Y ). ANS f X,Y (x, y) = J(x, y) = λα+β Γ(α)Γ(β) e λ(x+y) x α y β y (x+y) x (x+y) x = uv, y = u( v) f U,V (u, v) = λe λu (λu) α+β Γ(α + β) = x + y v α ( v) β Γ(α + β) Γ(α)Γ(β)

58 Joint Probability Distributions of Functions of Random Variables In many situations, the transformation of interest is not one-to-one. Suppose A has a partition of A, A,, A k, such that y = g (x, x ) and y = g (x, x ) is one-to-one transformation from in A i to B, for i =,,, k. The set A, which is maybe empty, satisfies P((X, X ) A ) =. Denote the ith inverse by x = h i (y, y ),x = h i (y, y ). J i (x, x ) denotes the Jacobian determinant in A. f Y Y (y, y ) = k f X,X (h i, h i ) J i (x, x ) i=

59 Joint Probability Distributions of Functions of Random Variables Example: Distribution of the Ratio of normal variables Let X and Y be independent N(,) random variables. Consider the tranformation U = X /Y and V = Y. This transformation is not one-to-one. Let A = {(x, y) : y = }, Let A = {(x, y) : y > }, A = {(x, y) : y } J = J = v. f U,V (u, v) = v π e (u +)v /, u (, ), v (, ) marginal pdf of U f U (u) = π(u + ).

60 Joint Probability Distributions of Functions of Random Variables When the joint density function of the n random variables X, X,..., X n is given and we want to compute the joint density function of Y, Y,..., Y n, where Y = g (X, X,..., X n ), Y = g (X, X,..., X n ),..., Y n = g n (X, X,..., X n ) Jacobian determinant J(x,, x n ) = at all points (x,, x n ). g g x g g x g n x x x g n x g x n g x n g n x n the equations y = g(x,, x n ), y = g (x,, x n ),,y n = g n (x,, x n ) have a unique solution, say, x = h (y,, y n ),, x n = h n (y,, y n ).

61 Joint Probability Distributions of Functions of Random Variables f Y,,Y n (y,, y n ) = f X,,X n (x,, x n ) J(x,, x n ) where x = h (y,, y n ),, x n = h n (y,, y n ). Example Let X, X,..., X n be independent and identically distributed exponential random variables with rate λ. Let Y i = X + + X i, i =,, n (a) Find the joint density function of Y,, Y n. (b) Use the result of part (a) to find the density of Y n.