Probability generating functionsEdexcel A-Level Further Maths: Mind map

Exam questions on Probability generating functions

1. The discrete random variable $X$ has $\mathrm{P}(X=0)=0.2$, $\mathrm{P}(X=1)=0.5$ and $\mathrm{P}(X=2)=0.3$. The probability generating function of $X$ is $\mathrm{G}_X(t)$.
   Use $\mathrm{G}_X(t)$ to find $\mathrm{Var}(X)$.2 marks
2. The random variable $X$ has a geometric distribution with parameter $p=\frac14$, so $X$ is the number of trials up to and including the first success, with $\mathrm{P}(X=r)=q^{r-1}p$ for $r=1,2,3,\ldots$ and $q=1-p$.
   Given that $\mathrm{G}_X'(t)=\dfrac{4}{(4-3t)^2}$, find $\mathrm{G}_X''(1)$ and hence show that $\mathrm{Var}(X)=12$.2 marks
3. The random variable $X$ has distribution $\mathrm{B}(3,0.4)$ and the random variable $Y$ has distribution $\mathrm{Po}(2)$. The variables $X$ and $Y$ are independent and $Z=X+Y$.
   Show, from the definition of a probability generating function, that $\mathrm{G}_X(t)=(0.6+0.4t)^3$.3 marks