Mean and variance of a CRVAQA A-Level Further Maths: Subtopic test

1. 1
   The continuous random variable $X$ has probability density function $f(x)=\frac{3x^2}{8}$ for $0\le x\le2$, and $f(x)=0$ otherwise.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$1$
   • B$\frac{12}{5}$
   • C$\frac32$
   • D$6$

   (b)

   Find $E(X^2)$.

   [1 mark]

   • A$\frac{12}{5}$
   • B$\frac32$
   • C$\frac94$
   • D$\frac{3}{20}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The random variable $X$ has mean $E(X)=5$ and variance $\text{Var}(X)=4$.

   (a)

   Find $E(4X+1)$.

   [1 mark]

   • A$20$
   • B$9$
   • C$24$
   • D$21$

   (b)

   Find $\text{Var}(4X+1)$.

   [1 mark]

   • A$16$
   • B$64$
   • C$17$
   • D$4$

   (c)

   The random variable $Y=2-3X$. Find $E(Y)$ and $\text{Var}(Y)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has probability density function $f(x)=\frac{2}{x^2}$ for $1\le x\le2$, and $f(x)=0$ otherwise.

   (a)

   Find $E(X)$.

   [3 marks]

   (b)

   Find $E(18X^{-3})$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The continuous random variable $X$ has probability density function $f(x)=\frac{3}{32}x(4-x)$ for $0\le x\le4$, and $f(x)=0$ otherwise.

   (a)

   Find $E(X)$, $E(X^2)$ and $\text{Var}(X)$.

   [6 marks]

   (b)

   (i) Find $E(5X^3)$.
   (ii) The random variable $Y=3-2X$. Using your answers to part (a), find $E(Y)$ and $\text{Var}(Y)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions