Two fair dice are rolled. Find the joint probability mass function of X and Y...

Tara Alvarado

Tara Alvarado

Answered

2021-12-18

Two fair dice are rolled. Find the joint probability mass function of X and Y when (a) X is the largest value obtained on any die and Y is the sum of the values; (b) X is the value on the first die and Y is the larger of the two values; (c) X is the smallest and Y is the largest value obtained on the dice.

Answer & Explanation

Stella Calderon

Stella Calderon

Expert

2021-12-19Added 35 answers

Define N1 and N2 as random variables that mark numbers obtained on the first and the second die. We know that N1 and N2 are independent and that N1,N2 
a) Here we have that X=max(N1,N2) and Y=N1+N2. Thus X{1,,6} and Y{2,,12}. Also we have that X<Y almost certainly. So take any k<l, where k and l are from the ranges given above. Consider event X=k,Y=l. That menas that the maximum value on any die is k and that the sum on both dice is l. Observe that the only possible pairs of (N1,N2) corresponding to that event are (k,l,k) and (l,k,k) if l<2k. If l=2k, the only possible pair s (k,k). Hence the required PMF is 
P(X=k,Y=l)={236,k<l<2k136,l=2k

rodclassique4r

rodclassique4r

Expert

2021-12-20Added 37 answers

Here we have that X=N1 and Y=max(N1,N2). Observe that both variables are in {1,,6} and that XY almost certainly. Take any kl from the range given above. We have that
P(X=k, Y=l)=P(Y=lX=k)P(X=k)
Suppose that k=l and that we are given X=k. In that case, N2 can be any number from the range 1,,k to obtain the required Y=l. Hence
P(Y=lX=k)P(X=k)=k616=k36
If k<l and we are giveen that X=k,N2 must be equal to l obtain Y=l. So, in that case
P(Y=lX=k)P(X=k)=1616=136
RizerMix

RizerMix

Expert

2021-12-29Added 437 answers

Here we have that X=min(N1,N2) and Y=max(N1,N2). We also have that XY almost certainy. So, take any kl
Suppose that k<l. In this case we have to have N1=k, N2=l or N1=l, N2=k. So, there are only two possibilities, hence
P(X=k,Y=l)=236
if k=l, the only possibility is (N1,N2)=(k,k), thus
P(X=k, Y=l)=136

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