Find the CDF of an rv X DUnif12nSolutionXUnif12n Let F denot

Find the CDF of an r.v. X DUnif(1,2,...,n).

X~Unif{1,2,..........,n}

Let F denote the cdf of X

Clearly, for u>=n , F(u)=1

and, for u<1 , f(u)=0

Now, For 1<=u<n

Let [.] denote the greatest integer function. Then, [u] is the greatest integer less than or equal to u.

F(u)=P(X<=u)=P(X<=[u])

=P(X=1 or 2 or ........ or [u])

=P(X=1) + P(X=2) + P(X=3) + ..........+P(X=[u])

=1/n + 1/n + .................. + 1/n ( [u] many times )

=[u]/n .............. for 1<=u<n