Find $\mathrm {E} (X), \mathrm {Var} (X)$. 

A sample of size $n$ is drawn from $\{1,2, \cdots , N \}$ with replacement. Let $X$ denote the minimum of the numbers drawn. Calculate
$(a)$ the  PMF of $X,$
$(b)$ $\mathrm E (X),$
$(c)$ $\mathrm {Var} (X),$
$(d)$ If $Y$ denotes the maximum of the numbers drawn$,$ calculate the joint PMF of $X$ and $Y$.


I have found 
$$P(X=k)= \frac {(N-k+1)^n - (N-k)^n} {N^n}.$$
But this isn't much helpful in calculating the expectation and variance of the random variable $X$. So how should I proceed in this regard? Please help me.
Thank you very much.
I have found the joint probability of the random variables $X$ and $Y$ as follows $:$
$$P(X=i,Y=j) =  \frac {(j-i+1)^n - 2(j-i)^n+(j-i-1)^n} {N^n}.$$ for $i=1,2,\cdots,N$ and $j=1,2,,\cdots,N$.
Is it correct? Please verify it.

Your answer for part (a) is correct.
For part (b), write your result from part (a) as
$$\mathsf P(X\gt k)=\frac{(N-k)^n}{N^n}$$
and calculate
$$
\mathsf E(X)=\sum_{k=0}^{N-1}\mathsf P(X\gt k)=\sum_{k=1}^N\frac{k^n}{N^n}\;,
$$
which can be evaluated using Faulhaber's formula. Approximating the sum by an integral for large $N$ yields the approximation $E(X)\approx\frac N{n+1}$.
For part (c), use the fact that the variance of the minimum is the variance of the maximum to transform to $j=N-k+1$, which leads to $j^n-(j-1)^n$ in the numerator; then use $j=(j-1)+1$ as appropriate to write the expectation of $j^2$ in terms of sums of powers of $j$.
Your answer for part (d) is correct; it was recently asked about in Calculating probability of High+Low = T on N dice with S sides.
