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im working on practice problems from the book and i have come across a question that i do not understand...

The joint probability distribution of X and Y is shown in the table:

        X
Y   1   2    3
2 0.10 0.15 0.20
4 0.30 0.15 0.10

a)Define the new random variable W = 2X + Y. Find P[ W = 8 ]?

  • In the book ive never had to define a "new variable" and google hasn't provided any answers either. By looking at it it looks like i have to find the probability of this new random variable. Why is it setting the W = 8? Any explanation would be great, thanks
Jovis13
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1 Answers1

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Note that statement: $$W=8$$ comes to the same as the statement: $$[X=2\text{ and }Y=4]\text{ or }[X=3\text{ and }Y=2]$$

The so called events "$X=2\text{ and }Y=4$" and "$X=2\text{ and }Y=4$" are disjoint, i.e. cannot occur both so you can find $P(W=8)$ on base of: $$P(W=8)=P(X=2\text{ and }Y=4)+P(X=3\text{ and }Y=2)$$ Now the table can be used. The setting $W=8$ is just coincidental. It can also be asked to find e.g. $P(W=4)$.

drhab
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