Question

4. Consider two random variables ? ∈ {0, 1, 2} and ? ∈ {1, 2} with marginal distributions ??(?) = 1 for ? = 0,1,2 ?? (?) = 1 for ? = 1,2

   (a) Come up with two di erent joint PMF-s that give these marginals
   (?*) Come up with a joint PMF that gives these marginals and guarantees that

   ?(? ?)=1
   (?*) Come up with a joint PMF that makes ? (? ̸= ? ) as small as possible

Answer 1

Let xeço, 1, 2} & y = $1,23 with. fx (2) = 3 x = 0, 1, 2 , foly) = 2 3 4 = 1,2 Let X&Y be independent .. fxymiy ] = fxxl fyly) joint pont of all fry/n,y) , tx121) O on 21 21+ Nt at ot てI TT 917 out wt wh fry) Ź and joint fry (4) pmf of x 2 ay also give the fxron) & fy CY) xron] X II at at on 2 lo Llwt wt wt fyld) P4x) = PCC, 12 HP31) +PC, 2) HP(1,27 +P/2,27 = + tot tett 3 I Pry >x) = 1]

سیمه - Sxy 11,Y) Z=x-Y) م о сох - we cod at o wt م - رN flund F-X = 2 TO 11-12 lž P(x + 4) = P(x-40) -P(2+0) - P(22)+1(2=-1) +P(z=1]. - ot to [ P(XFY ) = + 7