Proof and derivation of E(X) and Var(X)

The random variable $X$ has a binomial distribution with parameters $n$ and $p$.
1. Prove, from first principles, that $\mathrm { E } ( X ) = n p$.
2. Hence, given that $\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$, find, in terms of $n$ and $p$, an expression for $\operatorname { Var } ( X )$.
The mode, $m$, of $X$ is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
1. Use the first inequality to show that $$m \leqslant ( n + 1 ) p$$
2. Given that the second inequality results in $$m \geqslant ( n + 1 ) p - 1$$ deduce that the distribution $\mathrm { B } ( 10,0.65 )$ has one mode, and find the two values for the mode of the distribution $B ( 35,0.5 )$.
The random variable $Y$ has a binomial distribution with parameters 4000 and 0.00095 . Use a distributional approximation to estimate $\mathrm { P } ( Y \leqslant k )$, where $k$ denotes the mode of $Y$.
(3 marks)