Probability generating functionsEdexcel A-Level Further Maths: Subtopic test

1. 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 $\mathrm{G}_X(t)$.

   (a)

   Find $\mathrm{G}_X(t)$.

   [1 mark]

   • A$0.3+0.5t+0.2t^2$
   • B$0.5t+0.6t^2$
   • C$0.2+0.5t+0.3t^2$
   • D$0.2t+0.5t^2+0.3t^3$

   (b)

   Use $\mathrm{G}_X(t)$ to find $\mathrm{E}(X)$.

   [1 mark]

   • A$1.1$
   • B$1.7$
   • C$0.6$
   • D$1$

   (c)

   Use $\mathrm{G}_X(t)$ to find $\mathrm{Var}(X)$.

   [2 marks]

   Total for question 1: 4 marks

2. 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 $q=1-p$.

   (a)

   Find $\mathrm{G}_X(t)$.

   [1 mark]

   • A$\dfrac{3t}{4-t}$
   • B$\dfrac{t}{4+3t}$
   • C$\dfrac{1}{4-3t}$
   • D$\dfrac{t}{4-3t}$

   (b)

   Use $\mathrm{G}_X(t)$ to find $\mathrm{E}(X)$.

   [1 mark]

   • A$\frac14$
   • B$4$
   • C$12$
   • D$\frac34$

   (c)

   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]

   Total for question 2: 4 marks

3. 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 $Z=X+Y$.

   (a)

   Show, from the definition of a probability generating function, that $\mathrm{G}_X(t)=(0.6+0.4t)^3$.

   [3 marks]

   (b)

   The probability generating function of $Y$ is $\mathrm{G}_Y(t)=\mathrm{e}^{2(t-1)}$. Find the probability generating function of $Z$, and hence find $\mathrm{E}(Z)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A biased coin shows heads with probability $p=0.2$ and tails with probability $q=0.8$. The coin is tossed repeatedly. The random variable $X$ is the number of tosses up to and including the first head, with $\mathrm{G}_X(t)$ its probability generating function.

   (a)

   Show, from the definition, that $\mathrm{G}_X(t)=\dfrac{pt}{1-qt}$, stating the condition on $t$ for the series to converge. Hence prove that $\mathrm{E}(X)=\dfrac1p$.

   [6 marks]

   (b)

   The random variable $N$ is the number of tosses up to and including the third head. Find the probability generating function of $N$, and use $\mathrm{G}_X'(1)=5$ and $\mathrm{G}_X''(1)=40$ to find $\mathrm{E}(N)$ and $\mathrm{Var}(N)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions