Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / M / M4 / b

Expand the analytic factor around zero by the binomial series:Every coefficient is nonnegative. Adding $2/3$ to the constant coefficient gives probabilities $p_j$ with sum $g(1)=2/3+1/3=1$. Define $\mathrm{Pr}(T=j)=p_j$ for $j\ge 0$. This is a nonnegative integer-valued random variable and its probability generating function is exactly $g$. The power series has radius $14/5>1$, so the evaluations and derivatives used below are finite.

For a probability generating function, $\mathbb{E}T=g^{\prime }(1)$ and $\mathbb{E}[T(T-1)]=g^{\prime \prime }(1)$. HereThus $\mathbb{E}T=5/54$ and $\mathbb{E}[T(T-1)]=25/324$. Adding the first moment gives $\mathbb{E}{T}^2=55/324$, hence

Markov's inequality states that for a nonnegative random variable $Z$ and $a>0$,It follows immediately from $Z\ge a{1}_{\{Z\ge a\}}$ and monotonicity of expectation. A finite expectation makes the bound informative.

By the definition of the probability generating function,The evaluation at two is valid because two is inside the power-series radius $14/5$.

The event $T\ge 5$ is the same as $2^T\ge 32$. Apply Markov's inequality to the nonnegative random variable $2^T$ and use part (iv):The strict numerical comparison is enough even though the probabilistic inequality itself is non-strict.