probability of maximum of (x,y)

Question

I was studying for the maximum and minimum of joint probability

I have a question that " Let X and Y be independent each uniformly distributed on {1,2...,n}"

a) P(max(X,Y) = k) for 1 <=k <= n;

$$\sum_{i=0}^{k-1}P(X=k,Y=i) + \sum_{i=0}^{k-1}P(X=i,Y=k) + P(X=k, Y = k) $$

I come up with the answer like $$\sum_{i=0}^{k-1}P(X=k,Y=i) = \sum_{i=0}^{k-1}P(X=k)P(Y=i) = (1/n)*(k-1)*(1/n) $$

and $$P(X = k, Y = k) = (1/n) * (1/n) $$

Therefore $$ 2((k-1)/n^2)) + (1/n^2) $$

is this the right approach to translating summation to k-1 and multiply by (1/n) like that?

thank you so much

Answer 1

Same approach but slightly more concise notaions:

$P(max(X,Y)=k)=P(k,k)+2.P(k,\leq k-1)$ (by symmetry)

$=(P(k))^2+2.P(k).F(k-1)$ (by independence)

$=\frac{1}{n^2}+2\frac{k-1}{n^2}$