Two dice are tossed. Let X be the smaller number of points. Let Y be the larger number of points. If both dice show the same number, say, z points, then X = Y = z. (a) Find the joint probability mass function of (X, Y ). (b) Are X and Y independent? Explain. (c) Find the probability mass function of X. (d) If X = 2, what is the probability that Y = 5?
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Welcome to MSE. If you show us that you made an effort we will be happy to help you. – Jan 12 '18 at 10:50
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Let $X_1$ denote the value of the first dice, $X_2$ denote the value of the second dice. Then $X=\min(X_1,X_2),Y=\max(X_1,X_2)$. Then for $x\neq y$, $$P(X=x,Y=y)=P(X_1=x,X_2=y)+P(X_1=y,X_2=x)=2P(X_1=x)P(X_2=y)$$ and for $x=y$, $$P(X=x,Y=x)=P(X_1=x,X_2=x)=P(X_1=x)P(X_2=x)$$ noting $X_1$, $X_2$ are independent. You can easily find out the joint pmf from this.
If the joint could be written as product of two functions, one depending only on $x$ and the other only on $y$, then $X$ and $Y$ are independent, otherwise not.
Once you get the joint pmf, take it's sum over possible values of $Y$ to get the pmf of $X$.
QED
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Note $P(X=1,Y=2)=P(X_1=1,X_2=2)+P(X_1=2,X_2=1)=\frac1{36}+\frac1{36}=\frac1{18}$ – QED Jan 12 '18 at 11:06
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If i am taking P(X1=1,X2=2) why again should i consider P(X1=2,X2=1) – Satyaki Chatterjee Jan 12 '18 at 11:09
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Because $X$ is the maximum of $X_1,X_2$, while $Y$ is the minimum of the same. Now if only the values of $X$ and $Y$ are specified, $(X_1,X_2)$ can be either $(X,Y)$ or $(Y,X)$ – QED Jan 12 '18 at 11:11
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So sir are you assuming here greater value can come from any of the die? – Satyaki Chatterjee Jan 12 '18 at 11:18
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The following sheet shows the joint distribution of the min and the max of two rolls:
The min and the max are not independent:
For min=1 and max =3, the probability that the min is 1 is the sum of the numbers in the blue frame, $\approx 0.305$; the probability that the max is 3 is the sum of the numbers in the green frame, $\approx 0.138$; the product of these two numbers is $\approx 0.424$ and the probability that the min is 1 and the max is 3 is $\approx 0.555$.
The probability that the max is $5$ is the sum of the numbers in the red frame.
zoli
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