Joint Distribution of two or More Random Variables • Sometimes more than one measurement (r.v.) is taken on each member of the sample space. In cases like this there will be a few random variables defined on the same probability space and we would like to explore their joint distribution. • Joint behavior of 2 random variable (continuous or discrete), X and Y determined by their joint cumulative distribution function FX ,Y x, y P X x, Y y . • n – dimensional case FX1 ,..., X n x1 ,..., x n P X 1 x1 , , X n x n . week 7 1 Discrete case • Suppose X, Y are discrete random variables defined on the same probability space. • The joint probability mass function of 2 discrete random variables X and Y is the function pX,Y(x,y) defined for all pairs of real numbers x and y by p X ,Y x, y P X x and Y y • For a joint pmf pX,Y(x,y) we must have: pX,Y(x,y) ≥ 0 for all values of x,y and p x, y 1 X ,Y x y week 7 2 Example for illustration • Toss a coin 3 times. Define, X: number of heads on 1st toss, Y: total number of heads. • The sample space is Ω ={TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}. • We display the joint distribution of X and Y in the following table • Can we recover the probability mass function for X and Y from the joint table? • To find the probability mass function of X we sum the appropriate rows of the table of the joint probability function. • Similarly, to find the mass function for Y we sum the appropriate columns. week 7 3 Marginal Probability Function • The marginal probability mass function for X is p X x p X ,Y x, y y • The marginal probability mass function for Y is pY y p X ,Y x, y x • Case of several discrete random variables is analogous. • If X1,…,Xm are discrete random variables on the same sample space with joint probability function p X1 ,... X n x1 ,...xm P X 1 x1 ,..., X m xm The marginal probability function for X1 is p X1 x1 p X1 ,... X n x1,...xm x2 ,..., xm • The 2-dimentional marginal probability function for X1 and X2 is p X X x1 , x2 p X ,... X x1 , x2 , x3 ,..., xm 1 2 1 x3 ,..., xm n week 7 4 Example • Roll a die twice. Let X: number of 1’s and Y: total of the 2 die. There is no available form of the joint mass function for X, Y. We display the joint distribution of X and Y with the following table. • The marginal probability mass function of X and Y are • Find P(X ≤ 1 and Y ≤ 4) week 7 5 Independence of random variables • Recall the definition of random variable X: mapping from Ω to R such that : X x F for x R . • By definition of F, this implies that (X > 1.4) is and event and for the discrete case (X = 2) is an event. • In general X A : X A is an event for any set A that is formed by taking unions / complements / intersections of intervals from R. • Definition Random variables X and Y are independent if the events X A and Y B are independent. week 7 6 Theorem • Two discrete random variables X and Y with joint pmf pX,Y(x,y) and marginal mass function pX(x) and pY(y), are independent if and only if p X ,Y x, y p X x pY y • Proof: • Question: Back to the rolling die 2 times example, are X and Y independent? week 7 7 Conditional Probability on a joint discrete distribution • Given the joint pmf of X and Y, we want to find P X x and Y y P X x | Y y PY y and PY y | X x P X x and Y y P X x • Back to the example on slide 5, PY 2 | X 1 0 2 1 P Y 3 | X 1 36 10 5 36 PY 12 | X 1 0 • These 11 probabilities give the conditional pmf of Y given X = 1. week 7 8 Definition • For X, Y discrete random variables with joint pmf pX,Y(x,y) and marginal mass function pX(x) and pY(y). If x is a number such that pX(x) > 0, then the conditional pmf of Y given X = x is pY | X y | x pY | X y | X x • Is this a valid pmf? p X ,Y x, y p X x • Similarly, the conditional pmf of X given Y = y is p X |Y x | y p X |Y x | Y y p X ,Y x, y pY y • Note, from the above conditional pmf we get p X ,Y x, y p X |Y x | y pY y Summing both sides over all possible values of Y we get p X x p X ,Y x, y p X |Y x | y pY y y y This is an extremely useful application of the law of total probability. • Note: If X, Y are independent random variables then PX|Y(x|y) = PX(x). week 7 9 Example • Suppose we roll a fair die; whatever number comes up we toss a coin that many times. What is the distribution of the number of heads? Let X = number of heads, Y = number on die. We know that 1 pY y , y 1, 2, 3, 4, 5, 6 6 Want to find pX(x). • The conditional probability function of X given Y = y is given by y 1 p X |Y x | y x 2 2 for • By the Law of Total Probability we have x = 0, 1, …, y. y 1 1 p X x p X |Y x | y pY y 6 y y x x 2 6 y Possible values of x: 0,1,2,…,6. week 7 10 Example • Interested in the 1st and 2nd success in repeated Bernoulli trails with probability of success p on any trail. Let X = number of failures before 1st success and Y = number of failures before 2nd success (including those before the 1st success). • The marginal pmf of X and Y are for x = 0, 1, 2, …. p X x P X x pq x • and pY y PY y y 1 p 2 q y for y = 0, 1, 2, …. • The joint mass function, pX,Y(x,y) , where x, y are non-negative integers and x ≤ y is the probability that the 1st success occurred after x failure and the 2nd success after y – x more failures (after 1st success), e.g. p X ,Y 2,5 PFFSFFFS p 2 q 5 • The joint pmf is then given by p 2q y p X ,Y x, y 0 0 x y otherwise • Find the marginal probability functions of X, Y. Are X, Y independent? • What is the conditional probability function of X given Y = 5 week 7 11 Some Notes on Multiple Integration • Recall: simple integral f x dx cab be regarded as the limit as N ∞ of a the Riemann sum of the form N b f x x i i 1 i where ∆xi is the length of the ith subinterval of some partition of [a, b]. • By analogy, a double integral f x, y dA D where D is a finite region in the xy-plane, can be regarded as the limit as N ∞ of the Riemann sum of the form N N f x , y A f x , y x y i 1 i i i i 1 i i i i where Ai is the area of the ith rectangle in a decomposition of D into rectangles. week 7 12 Evaluation of Double Integrals • Idea – evaluate as an iterated integral. Restrict D now to a region such that any vertical line cuts its boundary in 2 points. So we can describe D by yL(x) ≤ y ≤ yU(x) and xL ≤ x ≤ xU • Theorem Let D be the subset of R2 defined by yL(x) ≤ y ≤ yU(x) and xL ≤ x ≤ xU If f(x,y) is continuous on D then D xU yU f x, y dA f x, y dy dx xL yL • Comment: When evaluating a double integral, the shape of the region or the form of f (x,y) may dictate that you interchange the role of x and y and integrate first w.r.t x and then y. week 7 13 Examples 1) Evaluate x ydA where D is the triangle with vertices (0,0), (2,0) and (0,1). D 2) Evaluate 2 2 xy x y dA where D is the region in the 1st quadrate bounded D by y = 0, y = x and x2 + y2 = 1. 3) Integrate f x, y xe x2 y over the set of points in the positive quadrate for which y > 2x. week 7 14 The Joint Distribution of two Continuous R.V’s • Definition Random variables X and Y are (jointly) continuous if there is a non-negative function fX,Y(x,y) such that P X , Y A f X ,Y x, y dxdy A for any “reasonable” 2-dimensional set A. • fX,Y(x,y) is called a joint density function for (X, Y). • In particular , if A = {(X, Y): X ≤ x, Y ≤ x}, the joint CDF of X,Y is FX ,Y x, y x y f X ,Y u, v du, dv • From Fundamental Theorem of Calculus we have 2 2 f X .Y x, y FX ,Y x, y FX ,Y x, y xy yx week 7 15 Properties of joint density function • f X ,Y x, y 0 for all x, y R • It’s integral over R2 is f X ,Y x, y dxdy 1 week 7 16 Example • Consider the following bivariate density function 12 2 x xy f X ,Y x, y 7 0 0 x 1, 0 y 1 otherwise • Check if it’s a valid density function. • Compute P(X > Y). week 7 17 Properties of Joint Distribution Function For random variables X, Y , FX,Y : R2 [0,1] given by FX,Y (x,y) = P(X ≤ x,Y ≤ x) lim FX ,Y x, y 0 x y lim FX ,Y x, y 1 x y • FX,Y (x,y) is non-decreasing in each variable i.e. FX ,Y x1 , y1 FX ,Y x2 , y 2 if x1 ≤ x2 and y1 ≤ y2 . • lim FX ,Y x, y FY y x and lim FX ,Y x, y FX x y week 7 18 Marginal Densities and Distribution Functions • The marginal (cumulative) distribution function of X is FX x P X x x f X ,Y u, y dydu • The marginal density of X is then f X x F x f X ,Y x, y dy ' X • Similarly the marginal density of Y is fY y f X ,Y x, y dx week 7 19 Example • Consider the following bivariate density function 6 xy2 f X ,Y x, y 0 0 x 1, 0 y 1 otherwise • Check if it is a valid density function. • Find the joint CDF of (X, Y) and compute P(X ≤ ½ ,Y ≤ ½ ). • Compute P(X ≤ 2 ,Y ≤ ½ ). • Find the marginal densities of X and Y. week 7 20 Generalization to higher dimensions Suppose X, Y, Z are jointly continuous random variables with density f(x,y,z), then • Marginal density of X is given by: f X x f x, y, z dydz • Marginal density of X, Y is given by : f X ,Y x, y f x, y, z dz week 7 21 Example Given the following joint CDF of X, Y FX ,Y x, y x 2 y y 2 x x 2 y 2 0 x 1, 0 y 1 • Find the joint density of X, Y. • Find the marginal densities of X and Y and identify them. week 7 22 Example Consider the joint density 2 e y f X ,Y x, y 0 0 x y otherwise where λ is a positive parameter. • Check if it is a valid density. • Find the marginal densities of X and Y and identify them. week 7 23