Sums of independent random variablesAQA A-Level Further Maths: Subtopic test

1. 1
   $X$ and $Y$ are independent random variables with $E(X)=12$, $\text{Var}(X)=9$, $E(Y)=7$ and $\text{Var}(Y)=16$.

   (a)

   Find $E(X+Y)$.

   [1 mark]

   • A$5$
   • B$84$
   • C$25$
   • D$19$

   (b)

   Find $\text{Var}(X+Y)$.

   [1 mark]

   • A$5$
   • B$7$
   • C$25$
   • D$19$

   (c)

   A student claims that the standard deviation of $X+Y$ is $3+4=7$. Show that this is wrong and state the correct value.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The times, in minutes, taken to prepare and to cook a dish are modelled by independent random variables $X$ and $Y$, with $E(X)=8.5$, $\text{Var}(X)=1.44$, $E(Y)=12.3$ and $\text{Var}(Y)=2.56$. The total time is $T=X+Y$.

   (a)

   Find $E(T)$.

   [1 mark]

   • A$3.8$
   • B$20.8$
   • C$4$
   • D$10.4$

   (b)

   Find $\text{Var}(T)$.

   [1 mark]

   • A$4$
   • B$20.8$
   • C$2$
   • D$2.8$

   (c)

   Find the standard deviation of $T$, and state the assumption that makes it valid to add the variances.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has probability density function $f(x)=\frac{x}{2}$ for $0\le x\le2$, and $f(x)=0$ otherwise. The discrete random variable $Y$ is the score when a fair six-sided die is rolled, so $E(Y)=\frac72$ and $\text{Var}(Y)=\frac{35}{12}$. $X$ and $Y$ are independent.

   (a)

   Find $E(X+Y)$.

   [3 marks]

   (b)

   Find $\text{Var}(X+Y)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A commuter's journey consists of a walk of $W$ minutes followed by a bus ride of $B$ minutes, where $W$ and $B$ are independent. $W$ has probability density function $f(w)=\frac16$ for $4\le w\le10$, and $0$ otherwise. $B$ has probability density function $g(b)=\frac{b}{8}$ for $0\le b\le4$, and $0$ otherwise. The total journey time is $T=W+B$.

   (a)

   Find $E(W)$ and $\text{Var}(W)$.

   [6 marks]

   (b)

   Find $E(T)$ and $\text{Var}(T)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions