Multivariate Distributions Marginal Distributions Conditional Distributions Lecture 4: Probability Distributions and Probability Densities - 2 Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics Istanbul˙ K¨ult¨ur University Multivariate Distributions Marginal Distributions Conditional Distributions Outline 1 Multivariate Distributions 2 Marginal Distributions 3 Conditional Distributions Multivariate Distributions Marginal Distributions Conditional Distributions Outline 1 Multivariate Distributions 2 Marginal Distributions 3 Conditional Distributions Multivariate Distributions Marginal Distributions Conditional Distributions In this section we shall concerned first with the bivariate case, that is, with situation where we are interested at the same time in a pair of random variables defined over a joint sample space that are both discrete. Later, we shall extend this discussions to the multivariate case, covering any finite number of random variables. If X and Y are discrete random variables, we write the probability that X will take on the value x and Y will take on the value y as P(X = x, Y = y). Thus, P(X = x, Y = y) is the probability of the intersection of the events X = x and Y = y.Asinthe univariate case, where we dealt with one random variable and could display the probabilities associated with all values of X by means of a table, we can now, in the bivariate case, display the probabilities associated with all pairs of the values of X and Y by mean of a table. Multivariate Distributions Marginal Distributions Conditional Distributions Example 1 Two caplets are selected at a random form a bottle containing three aspirin, two sedative, and four laxative caplets. If X and Y are, respectively, the numbers of the aspirin and sedative caplets included among the two caplets drawn from the bottle, find the probabilities associated with all possible pairs of values of X and Y . Solution. The possible pairs are (0, 0), (0, 1), (1, 0), (1, 1), (0, 2), and (2, 0). So we obtain the following probabilities: 3 2 4 6 P(X =0, Y =0)= 0 0 2 = , 9 36 2 3 2 4 8 P(X =0, Y =1)= 0 1 1 = , 9 36 2 3 2 4 12 P(X =1, Y =0)= 1 0 1 = , 9 36 2 Multivariate Distributions Marginal Distributions Conditional Distributions 3 2 4 6 P(X =1, Y =1)= 1 1 0 = , 9 36 2 3 2 4 1 P(X =0, Y =2)= 0 2 0 = , 9 36 2 3 2 4 3 P(X =2, Y =0)= 2 0 0 = . 9 36 2 Therefore, we have the following table: H HH x H 0 1 2 y HH 0 6/36 12/36 3/36 1 8/36 6/36 2 1/36 Multivariate Distributions Marginal Distributions Conditional Distributions Definition 2 If X and Y are discrete random variables, the function given buy f (x, y)=P(X = x, Y = y) for each pair of values (x, y) within the range of X and Y is called the joint probability distribution of X and Y . Theorem 3 A bivariate function can serve as the joint probability distribution of a pair of discrete random variables X and Y if and only if its values f (x, y) satisfy the conditions 1 f (x, y) ≥ 0 for each pair of values (x, y) within its domain; 2 x y f (x, y)=1, where the double summation extends over all possible pairs (x, y) within its domain. Multivariate Distributions Marginal Distributions Conditional Distributions Suppose that X canassumeanyoneofm values x1, x2, ··· , xm and Y canassumeanyoneofn values y1, y2, ··· , yn. Then the probability of the event that X = xj and Y = yk is given by P(X = xj , Y = yk )=f (xj , yk ). A joint probability function for X and Y can be represented by a joint probability table as in the following: H HH y H y1 y2 ··· yn Totals ↓ x HH x1 f (x1, y1) f (x1, y2) ··· f (x1, yn) g(x1) x2 f (x2, y1) f (x2, y2) ··· f (x2, yn) g(x2) . ··· . xm f (xm, y1) f (xm, y2) ··· f (xm, yn) g(xm) Totals → h(y1) h(y2) ··· h(yn) 1 Multivariate Distributions Marginal Distributions Conditional Distributions Example 4 Determine the value of k for which the function given by f (x, y)=kxy for x =1, 2, 3; y =1, 2, 3 can serve as a joint probability distribution. Solution. Substituting the various values of x and y,weget f (1, 1) = k, f (1, 2) = 2k, f (1, 3) = 3k, f (2, 1) = 2k, f (2, 2) = 4k, f (2, 3) = 6k, f (3, 1) = 3k, f (3, 2) = 6k, f (3, 3) = 9k. To satisfy the first condition of Theorem 3, the constant k must be nonnegative, and to satisfy the second condition k +2k +3k +2k +4k +6k +3k +6k +9k =1 so that 36k =1andk =1/36. Multivariate Distributions Marginal Distributions Conditional Distributions f (x, y)=kxy for x =1, 2, 3; y =1, 2, 3 The joint probability function for X and Y can be represented by a joint probability table as in the following: H HH y H 1 2 3 Totals ↓ x HH 1 k 2k 3k 6k 2 2k 4k 6k 12k 3 3k 6k 9k 18k Totals → 6k 12k 18k 36k =1 Multivariate Distributions Marginal Distributions Conditional Distributions Example 5 If the values of the joint probability distribution of X and Y are as shown in the table H H x HH 0 1 2 y HH 0 1/12 1/6 1/24 1 1/4 1/4 1/40 2 1/8 1/20 3 1/120 find (a) P(X =1, Y =2); (b) P(X =0, 1 ≤ Y < 3); (c) P(X + Y ≤ 1); (d) P(X > Y ). Multivariate Distributions Marginal Distributions Conditional Distributions H HH x H 0 1 2 y HH 0 1/12 1/6 1/24 1 1/4 1/4 1/40 2 1/8 1/20 3 1/120 , 1 (a) P(X =1 Y =2)= 20 ; , ≤ < , , 1 1 3 (b) P(X =0 1 Y 3) = f (0 1) + f (0 2) = 4 + 8 = 8 ; ≤ , , , 1 1 1 1 (c) P(X + Y 1) = f (0 0) + f (1 0) + f (0 1) = 12 + 6 + 4 = 2 ; > , , , 1 1 1 7 (d) P(X Y )=f (1 0) + f (2 0) + f (2 1) = 6 + 24 + 40 = 30 . Multivariate Distributions Marginal Distributions Conditional Distributions Example 6 If the joint probability distribution of X and Y is given by f (x, y)=c(x2 + y 2)forx = −1, 0, 1, 3, y = −1, 2, 3 find the value of c. Solution. Since H HH x H −1 0 1 3 Totals ↓ y HH −1 2c 1c 2c 10c 15c 2 5c 4c 5c 13c 27c 3 10c 9c 10c 18c 47c Totals → 17c 14c 17c 41c 89c =1 1 then we have that c = 89 . Multivariate Distributions Marginal Distributions Conditional Distributions Example 7 Show that there is no value of k for which f (x, y)=ky(2y − x)forx =0, 3, y =0, 1, 2 can serve as the joint probability distribution of two random variables. Solution. Since H HH y H 0 1 2 Totals ↓ x HH 0 0 2k 8k 10k 3 0 −k 2k k Totals → 0 k 10k 11k =1 then we find that k =1/11.Butinthiscase,f (3, 1) differs in sign from all other terms. Multivariate Distributions Marginal Distributions Conditional Distributions Example 8 Suppose that we roll a pair of balanced dice and X is the number of dice that come up 1, and Y is the number of dice that come up 4, 5, or 6. (a) Construct a table showing the values X and Y associated with each of the 36 equally likely points of the sample space. (b) Construct a table showing the values the joint probability distribution of X and Y . Multivariate Distributions Marginal Distributions Conditional Distributions Solution. (a) If X is the number of dice that come up 1, and Y is the number of dice that come up 4, 5, or 6, then we have PP PP Roll 2 PP 1 2 3 4 5 6 Roll 1 PPP 1 (2,0) (1,0) (1,0) (1,1) (1,1) (1,1) 2 (1,0) (0,0) (0,0) (0,1) (0,1) (0,1) 3 (1,0) (0,0) (0,0) (0,1) (0,1) (0,1) 4 (1,1) (0,1) (0,1) (0,2) (0,2) (0,2) 5 (1,1) (0,1) (0,1) (0,2) (0,2) (0,2) 6 (1,1) (0,1) (0,1) (0,2) (0,2) (0,2) (b) Then, we simply count the number of times we have each of the possible (x, y) values, and divide by 36. x 0 0 0 1 1 2 y 0 1 2 0 1 0 Prob. 4/36 12/36 9/36 4/36 6/36 1/36 Multivariate Distributions Marginal Distributions Conditional Distributions Definition 9 If X and Y are discrete random variables, the function given by F (x, y)=P(X ≤ x, Y ≤ y)= f (s, t)for −∞< x, y < ∞ s≤x t≤y where f (s, t) is the value of the joint probability distribution of X and Y at (s, t), is called the joint distribution function,orthe joint cumulative distribution, of X and Y . Theorem 10 If F (x, y) is the value of the joint distribution function of two discrete random variables X and Y at (x, y),then (a) F (−∞, −∞)=0; (b) F (∞, ∞)=1; (c) if a < bandc< d, then F(a, c) ≤ F (b, d).