Mean and variance of continuous random variablesEdexcel A-Level Further Maths: Subtopic test

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

   (a)

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

   [1 mark]

   • A$1$
   • B$\frac{3}{2}$
   • C$\frac{12}{5}$
   • D$\frac{3}{4}$

   (b)

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

   [1 mark]

   • A$\frac{12}{5}$
   • B$\frac{9}{4}$
   • C$\frac{93}{20}$
   • D$\frac{3}{20}$

   (c)

   Find $\mathrm{E}\!\left(\frac{1}{X}\right)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $Y$ has probability density function $f(y)=\frac{y}{2}$ for $0\le y\le 2$, and $f(y)=0$ otherwise.

   (a)

   Find the mode of $Y$.

   [1 mark]

   • A$2$
   • B$\frac{4}{3}$
   • C$\sqrt2$
   • D$0$

   (b)

   Find the median of $Y$.

   [1 mark]

   • A$1$
   • B$\frac{4}{3}$
   • C$\sqrt2$
   • D$2$

   (c)

   Calculate $\mathrm{E}(Y)$. Using this value, together with the mode and median of $Y$, describe the skewness of the distribution, justifying your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has probability density function $f(x)=\frac{3}{x^4}$ for $x\ge 1$, and $f(x)=0$ otherwise.

   (a)

   Find the 90th percentile of $X$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A rail operator models the delay, $D$ minutes, of a particular train using the probability density function $f(d)=\frac{10-d}{50}$ for $0\le d\le 10$, and $f(d)=0$ otherwise.

   (a)

   (i) Find the median of $D$, giving your answer to 3 significant figures.
   (ii) Given that $\mathrm{E}(D)=\frac{10}{3}$, describe the skewness of the distribution, justifying your answer.

   [6 marks]

   (b)

   A random sample of 200 delays has mean 3.4 minutes and variance 5.8 minutes$^2$, and the longest delay in the sample is 12.5 minutes. Calculate $\mathrm{Var}(D)$ under the model and evaluate the suitability of the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions