Further Mathematics 9231: Probability generating function Topical Past Paper Questions

The probability generating function $G_{X}(t)$ of a random variable $X$ is given by $$ G_{X}(t)=\tfrac{1}{6}(t+1)(t^{2}+2). $$ (a) Use $ G_{X}(t) $ to find $ \mathrm{E}(X) $ and $ \mathrm{Var}(X) $. [Unlabeled] (b) The random variable Y is the sum of three independent observation…

The random variable $X$ takes values 1 and 2 with probabilities $\frac{2}{5}$ and $\frac{3}{5}$ respectively. (a) Write down the probability generating function $ G_{X}(t) $ of X. [1] The random variable Y is the sum of four independent observations of X. (b) Find the probability…

Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is $ \frac{1}{3} $ and the third coin is biased so that the probability of obtaining a head is $ \frac{1}{5} $. The random variable X is the number of heads that Nikita obtain…