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.

Stella Calderon

Expert

Define ${N}_{1}$ and ${N}_{2}$ as random variables that mark numbers obtained on the first and the second die. We know that ${N}_{1}$ and ${N}_{2}$ are independent and that ${N}_{1},{N}_{2}$
a) Here we have that $X=max\left({N}_{1},{N}_{2}\right)$ and $Y={N}_{1}+{N}_{2}$. Thus $X\in \left\{1,\dots ,6\right\}$ and $Y\in \left\{2,\dots ,12\right\}$. Also we have that X<Y almost certainly. So take any $k, 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 $\left({N}_{1},{N}_{2}\right)$ corresponding to that event are $\left(k,l,-k\right)$ and $\left(l,-k,k\right)$ if $l<2k$. If $l=2k$, the only possible pair s (k,k). Hence the required PMF is
$P\left(X=k,Y=l\right)=\left\{\begin{array}{ll}\frac{2}{36}& ,k

rodclassique4r

Expert

Here we have that $X={N}_{1}$ and $Y=max\left({N}_{1},{N}_{2}\right)$. Observe that both variables are in $\left\{1,\dots ,6\right\}$ and that $X\le Y$ almost certainly. Take any $k\le l$ from the range given above. We have that

Suppose that $k=l$ and that we are given $X=k$. In that case, ${N}_{2}$ can be any number from the range $1,\dots ,k$ to obtain the required $Y=l$. Hence
$P\left(Y=l\mid X=k\right)P\left(X=k\right)=\frac{k}{6}\cdot \frac{16}{=}\frac{k}{36}$
If $k and we are giveen that $X=k,{N}_{2}$ must be equal to l obtain $Y=l$. So, in that case
$P\left(Y=l\mid X=k\right)P\left(X=k\right)=\frac{16}{\cdot }\frac{16}{=}\frac{1}{36}$

RizerMix

Expert

Here we have that $X=min\left({N}_{1},{N}_{2}\right)$ and $Y=max\left({N}_{1},{N}_{2}\right)$. We also have that $X\le Y$ almost certainy. So, take any $k\le l$
Suppose that k<l. In this case we have to have or . So, there are only two possibilities, hence
$P\left(X=k,Y=l\right)=\frac{2}{36}$
if k=l, the only possibility is $\left({N}_{1},{N}_{2}\right)=\left(k,k\right)$, thus

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