Question

If x~Uniform{1, 2, ....12}

and Y = gcd(x, 6)

What is PMF of Y?

Answer 1

x	p(X=x)	Y=gcd(x, 6)
1	1/12	1
2	1/12	2
3	1/12	3
4	1/12	2
5	1/12	1
6	1/12	6
7	1/12	1
8	1/12	2
9	1/12	3
10	1/12	2
11	1/12	1
12	1/12	6

Y is taking value 1,2,3,and 6

p(Y=1) = p(x such that gcd(x,6)=1) = p(x=1)+p(x=5)+p(x=7)+p(x=11)=4/12

p(Y=2) = p(x such that gcd(x,6)=2) = p(x=2) +p(x=4)+p(x=8)+p(x=10)=4/12

p(Y=3) = p(x such that gcd(x,6)=3) = p(x=3)+p(x=9)=2/12

p(Y=6) = p(x such that gcd(x,6)=6) = p(x=6)+p(x=12)=2/12

PMF of Y

PMF of Y
y	p(Y=y)
1	4/12
2	4/12
3	2/12
6	2/12

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