Section 5.1: Joint Distribution of RandomVariables (Only Discrete Case)1Concepts and Formulae:• Let X and Y be discrete random variables.The joint PMF of X and Y is defined byp(x, y) = P (X = x, Y = y).• Let A be any set consisting of pairs (x, y) val-ues. Then,P [(X, Y ) ∈ A] =XX(x,y)∈Ap(x, y).• The marginal PMF of X ispX(x) =Xyp(x, y)and the marginal PMF of Y ispY(y) =Xxp(x, y).• X and Y are independent if for all (x, y)p(x, y) = pX(x)pY(y).• The conditional PMF of X given Y ispX|Y(x|y) =p(x, y)pY(y)and the conditional PMF of Y given X ispY |X(y|x) =p(x, y)pX(x).2First example of Section 5.1. Examples 5.1 and5.2 on textbook. The (joint) PMF of X and Y isyx 0 100 200 Total100 0.2 0.1 0.2 0.5250 0.05 0.15 0.30 0.5Total 0.25 0.25 0.5 1•P (Y ≥ 100) =0.1 + 0.2 + 0.15 + 0.30=0.75.P (X + Y ≤ 300) =0.2 + 0.1 + 0.2 + 0.05=0.55.• The marginal PMF of X isx 100 250pX(x) 0.5 0.5and the marginal PMF of Y isy 0 100 200pY(y) 0.25 0.25 0.53• Are X and Y independent?Answer: sincep(100, 0) = 0.2 6= pX(100)pY(0) = 0.5 × 0.25X and Y are not independent.• The conditional PMF of X given Y = 0 isx 100 250pX|Y =0(x) 0.8 0.2• The conditional PMF of X given Y = 100 isx 100 250pX|Y =100(x) 0.4 0.6• The conditional PMF of X given Y = 200 isx 100 250pX|Y =200(x) 0.4 0.64Second example of Section 5.1. The (joint) PMFof X and Y isyx 1 2 3 Total1 0.05 0.1 0.1 0.252 0.05 0.1 0.1 0.253 0.1 0.2 0.2 0.5Total 0.2 0.4 0.4 1• Are X and Y independent.Answer: We need to check whetherp(x, y) = pX(x)pY(y)for all pairs. Since all are correct, we concludeX and Y are independent.• If we assume X and Y are independent, thenwe can recover the joint PMF by marginalPMF.• If X and Y are independent, thenpX|Y(x) = pX(x)andPY |X(y) = pY(y).5Third example of Section 5.1. Suppose a bag has6 boxes, three of them containing 3 bullets, two ofthem containing 4 bullets, and one of them con-taining 5 bullets. A person is asked to randomlychoose a box and then shoot a target. Supposewith 60% for this person the bullet will hit thetarget. Let X be the number of bullets hittingthe target and Y be the number of bullets theperson has. Compute the joint PMF of X and Y .Answer: Clearly, the marginal PMF of Y isy 3 4 5pY(y)121316The conditional PMF of X given Y = y isBin(y, 0.6).Therefore, we havepX|Y =y(x) =³yx´0.6x0.4y−x.Note thatp(x, y) = pX|Y =y(x)pY(y)for x = 0, 1, · · · , y.6We have the following (joint) PMF tableyx 3 4 50 0.032 0.0085 0.00171 0.144 0.0512 0.01282 0.216 0.1152 0.03943 0.108 0.1152 0.05764 0 0.0432 0.04325 0 0 0.0130Q: what is the marginal PMF of