Further Mathematics: Statistics and Probabilities Topical Past Paper Questions

The continuous random variable X has probability density function f given by $$ f(x)=\left\{\begin{aligned}\frac{4x^{3}}{a^{4}}&\quad&0\leq x\leq a\\ 0&\quad&otherwise\end{aligned}\right. $$ where $a$ is a positive constant. To estimate the value of $a$, a random sample of $n$ ob…

[Maximum mark: 22] The discrete random variable X has the following probability distribution, where $ \theta $ is a constant. (a) State the range of possible values of $ \theta $. (b) Show that $ \mathrm{E}(X^{n})=2^{n}+2\theta(1-3\times2^{n-1}) $. In order to estimate $\theta$,…

[Maximum mark: 22] The discrete random variable X follows a geometric distribution $ \text{Geo}(p) $ where $$ \mathrm{P}\left(X=x\right)=\left\{\begin{aligned}&pq^{x-1},for x=1,2\ldots\\&0,otherwise\end{aligned}\right.. $$ (a) (i) Show that the probability generating function of…