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Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 4 / c / iii

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Oxford Undated Prelims Mathematics III resit 4 c
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iii
Summing over the last digit yields the marginal probability mass function
pY​(y)=(1−q)q10y∑x=09​qx=(1−q10)q10y,y≥0.
(1)
Together with the other marginal probability mass function, this gives
pX,Y​(x,y)=pX​(x)pY​(y).​
(2)
The factorization holds on the whole support and with zeros elsewhere, proving that X and Y are independent. The conditional distribution in ii likewise does not depend on the permissible value x.

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