Coursework questions

Show that an alternative expression for the variance of a random variable is given by

[3]

Let $R\sim \mathrm{Bin}(n,\theta )$ be a Binomial random variable.

Derive formulas for $\mathrm{E}[R]$ and $\mathrm{Var}(R)$.

[4]

Suppose that the average number of cars abandoned weekly on a certain highway is $2.2$. Assuming that this is a rare event find probability that there will be

1. (a)

   no abandoned cars in the next week;

2. (b)

   at least $2$ abandoned cars in the next week.

[3]