Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / M (Probability)

Uniform area distribution on the unit disk givesThe Jacobian determinant has absolute determinant $r$, soIntegrating out the other coordinate givesThe joint probability density function factors into these two marginal distribution, proving independence of $R$ and $\Theta$. FinallyThe angle at $R=0$ can be defined arbitrarily, since that event has probability zero.

An absorbing state $i$ has $p_{ii}=1$; once reached, the chain never leaves it. For a nonabsorbing initial state, condition on its first transition and use the Markov property:Here $h_1=1$ and $h_4=0$. The other equations areSolving givesFor absorption into either state one or four, let $e_i$ be the expected number of steps, with $e_1=e_4=0$. first-step analysis givesHence $e_2=2$ andThe expectation is finite because each transient two-step cycle has probability at most $(2/3)(1/4)=1/6$ of remaining transient; the survival tail is geometrically bounded. This also justifies solving the expectation equations without assuming finiteness in advance.

The moment-generating function is $M_X(t)=\mathbb{E}{e}^{tX}$ on the real values where this expectation is finite, normally including a neighborhood of zero. Direct substitution giveswhenever the right side is finite.

For the given Gamma distribution and $t<\lambda$,using the Gamma function after the scaling $u=(\lambda -t)x$. At $t\ge \lambda$ the positive integral diverges.

independence multiplies moment-generating functions. ThusUniqueness of a moment-generating function on a neighborhood of zero identifieswhere $\lambda$ is the rate parameter, as in the printed probability density function.