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.

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Stella Calderon · Accepted Answer

Define&amp;nbsp;N1&amp;nbsp;and&amp;nbsp;N2&amp;nbsp;as random variables that mark numbers obtained on the first and the second die. We know that&amp;nbsp;N1&amp;nbsp;and&amp;nbsp;N2&amp;nbsp;are independent and that&amp;nbsp;N1,N2&amp;nbsp;a) Here we have that&amp;nbsp;X=max(N1,N2)&amp;nbsp;and&amp;nbsp;Y=N1+N2. Thus&amp;nbsp;X∈{1,…,6}&amp;nbsp;and&amp;nbsp;Y∈{2,…,12}. Also we have that X&amp;lt;Y almost certainly. So take any&amp;nbsp;k&amp;lt;l, where k and l are from the ranges given above. Consider event&amp;nbsp;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&amp;nbsp;(N1,N2)&amp;nbsp;corresponding to that event are&amp;nbsp;(k,l,−k)&amp;nbsp;and&amp;nbsp;(l,−k,k)&amp;nbsp;if&amp;nbsp;l&amp;lt;2k. If&amp;nbsp;l=2k, the only possible pair s (k,k). Hence the required PMF is&amp;nbsp;P(X=k,Y=l)={236,k&amp;lt;l&amp;lt;2k136,l=2k

rodclassique4r · Accepted Answer

Here we have that X=N1 and Y=max(N1,N2). Observe that both variables are in {1,…,6} and that X≤Y almost certainly. Take any k≤l from the range given above. We have that  P(X=k, Y=l)=P(Y=l∣X=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=l∣X=k)P(X=k)=k6⋅16=k36  If k&amp;lt;l and we are giveen that X=k,N2 must be equal to l obtain Y=l. So, in that case  P(Y=l∣X=k)P(X=k)=16⋅16=136

RizerMix · Accepted Answer

Here we have that X=min(N1,N2) and Y=max(N1,N2). We also have that X≤Y almost certainy. So, take any k≤l Suppose that k&amp;lt;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

Two fair dice are rolled. Find the joint probability mass fu

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