Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 7 / b

The independent random variables have a product joint probability mass function. For integers $r,t\ge 0$,It is zero for all other pairs. The factors are the respective Poisson distribution masses, with the product justified by independence.

By the definition of a probability generating function and the power series for the exponential function,The series is absolutely convergent for every finite real $s$; in particular the expression is finite for all $s>1$, as needed for the later tail estimate.

For $Z=2(X+Y)+1$, independence givesThe initial factor $s$ accounts for the shift by one; substituting $s^2$ accounts for doubling the random variable. Its support consists of positive odd integers.

Conditioning on $Z=z$ has positive probability exactly when $z=2r+1$ for an integer $r\ge 0$. This is the same event as $X+Y=r$. The addition of independent Poisson random variables gives a Poisson distribution with parameter $\lambda +\mu$ for the sum. Therefore, for $0\le t\le r$,Thus $X\mid \{Z=2r+1\}\sim \mathrm{Binomial}(r,\lambda /(\lambda +\mu ))$. The case $r=0$ is a constant zero random variable. For even, nonpositive, or noninteger $z$, the conditioning event has zero probability, so an event-conditional probability distribution is not specified by the quotient formula. This is a two-component case of Poisson-multinomial conditioning.