<s>[INST] What is $P(\min\{X, Y\} = 1)$? If $y=1,2,3$ and $x=0,1,2$ where $P(X=x, Y=y) = k(2x+3y)$
I need to find $P(\min\{X, Y\} = 1)$.
I thought I need to use that the CMF of the minimum is $1-P(X)$, and maybe to find k by doing derivative on the equation and to sum it up to 1?
would love any direction on this.
 [/INST] Given that $P(X=x, Y=y) = k(2x+3y)$
$\sum_{(x,y)}P(X=x,Y=y) = k(0+3) +k(2 + 3) + k(4 +3) + k(0+6) + k(2+6) + k(4+6) + k(0+9) + k(2+9) + k(4+9) = 72k = 1$
Hence $k = \frac{1}{72}$
Now, $P(\min(X,Y)=1) = P(X=1,Y=2,3) + P(X=2, Y=1)$
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