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Random Number Sets

1. Choose a number at random from the set of numbers from the set of numbers {1,2,3,4,5}. Now choose a number from the subset {1,...,X). Call this second number Y.
a) Find the joint p.m.f. of X and Y
b) Find the conditional mass function of X given that Y = i. Do it for i = 1,2,3,4,5
c) Are X and Y independent? Why?

Solution Preview

a) P(X=1,Y=1)=1/5, P(X=1,Y=i)=0, i=2,3,4,5;
P(X=2,Y=1)=1/10, P(X=2,Y=2)=1/10, P(X=2,Y=i)=0,i=3,4,5;
P(X=3,Y=1)=1/15=P(X=3,Y=2)=P(X=3,Y=3),
P(X=3,Y=4)=P(x=3,Y=5)=0;
P(X=4,Y=1)=1/20=P(X=4,Y=2)=P(X=4,Y=3)=P(X=4,Y=4) P(x=3,Y=5)=0;
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Solution Summary

1. Choose a number at random from the set of numbers from the set of numbers {1,2,3,4,5}. Now choose a number from the subset {1,...,X). Call this second number Y.
a) Find the joint p.m.f. of X and Y
b) Find the conditional mass function of X given that Y = i. Do it for i = 1,2,3,4,5
c) Are X and Y independent? Why?

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