Distributions and expectation of DRVsAQA A-Level Further Maths: Subtopic test

1. 1
   The discrete random variable $X$ has probability distribution $P(X=x)=kx$ for $x=1,2,3,4$, and $P(X=x)=0$ otherwise, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac14$
   • B$\frac1{30}$
   • C$\frac1{24}$
   • D$\frac1{10}$

   (b)

   Find $P(X\ge3)$.

   [1 mark]

   • A$0.4$
   • B$0.3$
   • C$0.7$
   • D$0.9$

   (c)

   Find $E(X)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The discrete random variable $X$ has $P(X=1)=0.1$, $P(X=2)=0.3$, $P(X=3)=0.4$ and $P(X=4)=0.2$.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$2.5$
   • B$2.7$
   • C$3$
   • D$8.1$

   (b)

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

   [1 mark]

   • A$0.81$
   • B$8.1$
   • C$0.9$
   • D$5.4$

   (c)

   State the mode of $X$ and find the median of $X$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $X$ takes the values $1,2,3,4$ with $P(X=1)=0.2$, $P(X=2)=p$, $P(X=3)=q$ and $P(X=4)=0.3$, where $p$ and $q$ are constants. It is given that $E(X)=2.6$.

   (a)

   Find the values of $p$ and $q$.

   [3 marks]

   (b)

   Find the standard deviation of $X$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The discrete random variable $X$ has probability distribution $P(X=x)=\frac{k}{x}$ for $x=1,2,3,4$, and $P(X=x)=0$ otherwise, where $k$ is a constant.

   (a)

   (i) Show that $k=\frac{12}{25}$. (ii) Find $E(X)$. (iii) Find the median of $X$.

   [6 marks]

   (b)

   Find the variance and standard deviation of $X$, and hence find the probability that $X$ lies within one standard deviation of $E(X)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions