Past exam of the options II course of the University of Oxford / 2011 / M / M4 / b

The binomial series gives, for $|z|<14/5$,All coefficients are nonnegative, and $g(1)=2/3+1/3=1$. Thus defineThese numbers sum to $1$, so they define a probability distribution on the nonnegative integers with the desired probability generating function.

Differentiate the probability generating function:ThereforeUsing the relation between the second factorial moment and variance,

For a nonnegative random variable $Z$ and $u>0$, Markov inequality statesIt follows by taking expectations in $Z\ge u{1}_{\{Z\ge u\}}$. A finite expectation makes the bound informative.

Since $2$ lies inside the radius $14/5$ of the probability generating function, its nonnegative series converges at $2$, and

Apply Markov inequality to the nonnegative random variable $2^T$. On $\{T\ge 5\}$ it is at least $32$, soThis exponential bound is substantially stronger here than applying Markov inequality directly to $T$.