# Two independent uniform distribution random variables

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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$\rm\bf (d)$ 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?