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Let $X$ and $Y$ be independent, each uniformly distributed on {1,2,...,n}. Find:

a) $P(X=Y)$; b)$P(X < Y)$; c)$P(X>Y)$

d) $P(\max(X,Y)=k) \text{ for } 1\le k \le n$

e) $P(\min(X,Y)=k) \text{ for } 1\le k \le n$

f) $P(X+Y=k)$ for $2\le k \le 2n$

I could do part a,b,c by using symmetry but not sure how to approach the rest of the problem

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up vote 1 down vote accepted

A gentle guide:

$\rm\bf (d)$ Count the number of pairs $(i,j)$ with one of $i,j$ equal to $k$ and the other anything less than or equal to $k$. Make sure you don't count $(k,k)$ twice. Now divide by $n^2$. Why does this work?

$\rm\bf (e)$ Count the no. of pairs $(i,j)$ with one of them equal to $k$ and the other anything greater than or equal to $k$ - and don't doublecount $(k,k)$. Divide by $n^2$. Again, answer: why does this work?

$\rm\bf (f)$ Now, answer: what kind of pairs $(i,j)$ should you count here? 1. What are the possible values that the $i$ component can take on, and 2. given a valid $i$ component, how many possible values of $j$ are there such that $(i,j)$ is counted?

If you don't attempt to answer "why does this work" in each then you aren't giving it your all and you won't be able to do further questions like this on your own.

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