Further Statistics 1: Probability generating functionsEdexcel A-Level Further Maths: Topic test

1. 1
   The discrete random variable $X$ takes the values $1$, $2$ and $3$ and has probability generating function $\mathrm{G}_X(t)=kt(1+t)^2$, where $k$ is a constant.

   (a)

   What is the value of $k$?

   [1 mark]

   • A$\frac18$
   • B$\frac14$
   • C$\frac13$
   • D$\frac12$

   (b)

   What is $\mathrm{P}(X=2)$?

   [1 mark]

   • A$\frac14$
   • B$\frac13$
   • C$\frac12$
   • D$\frac34$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A quality inspector tests components one at a time until the first faulty one is found. Each component is faulty with probability $\frac13$, independently of the others. The random variable $X$ is the number of components tested.

   (a)

   What is $\mathrm{P}(X=3)$?

   [1 mark]

   • A$\frac{4}{27}$
   • B$\frac{8}{27}$
   • C$\frac29$
   • D$\frac{1}{27}$

   (b)

   What is $\mathrm{Var}(X)$?

   [1 mark]

   • A$2$
   • B$3$
   • C$9$
   • D$6$

   (c)

   Show that the probability generating function of $X$ is $\mathrm{G}_X(t)=\dfrac{t}{3-2t}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The independent random variables $X$ and $Y$ are the numbers of faulty items produced in one hour by two machines. $X\sim\mathrm{Po}(1.5)$ and $Y\sim\mathrm{Po}(2.5)$. The random variable $S=X+Y$.

   (a)

   Prove that the probability generating function of $X$ is $\mathrm{G}_X(t)=\mathrm{e}^{1.5(t-1)}$.

   [3 marks]

   (b)

   Find the probability generating function of $S$, identify the distribution of $S$ and hence find $\mathrm{P}(S=3)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The random variable $X$ is the number of trials needed to obtain the second success in a sequence of independent trials, each with probability of success $\frac14$. The probability generating function of $X$ is $\mathrm{G}_X(t)=\dfrac{t^2}{(4-3t)^2}$.

   (a)

   Use $\mathrm{G}_X(t)$ to show that $\mathrm{E}(X)=8$ and $\mathrm{Var}(X)=24$.

   [6 marks]

   (b)

   Two such sequences of trials are carried out independently and $Y$ is the total number of trials needed. Find the probability generating function of $Y$, state $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$, and use a binomial expansion to find $\mathrm{P}(Y=5)$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A farmer sows $5$ seeds. Each seed germinates with probability $0.3$, independently of the others. The random variable $X$ is the number of seeds that germinate.

   (a)

   Which expression is the probability generating function of $X$?

   [1 mark]

   • A$(0.3+0.7t)^5$
   • B$5(0.7+0.3t)$
   • C$(0.7+0.3t)^5$
   • D$0.7+0.3t^5$

   (b)

   What is $\mathrm{E}(X)$?

   [1 mark]

   • A$3.5$
   • B$1.5$
   • C$1.05$
   • D$5$

   (c)

   Find the coefficient of $t^2$ in the expansion of $\mathrm{G}_X(t)$ and state what it represents.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The discrete random variable $N$ takes the values $0,1,2,\dots$ and has probability generating function $\mathrm{G}_N(t)=\dfrac{1}{2-t}$ for $|t|<2$.

   (a)

   What is $\mathrm{P}(N=2)$?

   [1 mark]

   • A$\frac14$
   • B$\frac12$
   • C$\frac1{16}$
   • D$\frac18$

   (b)

   What is $\mathrm{E}(N)$?

   [1 mark]

   • A$1$
   • B$2$
   • C$\frac12$
   • D$\frac14$

   (c)

   Find $\mathrm{Var}(N)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The independent random variables $X$ and $Y$ are the numbers of defective items in two batches. Batch one has $8$ items and batch two has $12$ items. Each item in either batch is defective with probability $\frac14$, independently. The random variable $S=X+Y$.

   (a)

   Show that $S\sim\mathrm{B}\left(20,\frac14\right)$ by using probability generating functions.

   [3 marks]

   (b)

   Use $\mathrm{G}_S(t)$ to find $\mathrm{E}(S)$ and $\mathrm{Var}(S)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The number of emails an employee receives in one hour has a Poisson distribution with parameter $\lambda$, so the number $N$ has probability generating function $\mathrm{G}_N(t)=\mathrm{e}^{\lambda(t-1)}$. Two employees receive emails independently. In one hour the first receives $X\sim\mathrm{Po}(1.2)$ emails and the second receives $Y\sim\mathrm{Po}(1.8)$ emails. The total is $T=X+Y$.

   (a)

   Use differentiation of $\mathrm{G}_N(t)$ to prove that $\mathrm{E}(N)=\mathrm{Var}(N)=\lambda$.

   [6 marks]

   (b)

   Find the probability generating function of $T$ and hence find $\mathrm{P}(T=0)$ and $\mathrm{P}(T\geq2)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions