Question 1 - A-Level Maths

CAIE S2 2023 June — Question 1 3 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2023
Session	June
Marks	3
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Continuous Probability Distributions and Random Variables
Type	Find expectation E(X)
Difficulty	Moderate -0.8 This is a straightforward application of the expectation formula E(X) = ∫xf(x)dx with a simple polynomial pdf. The integration is routine (integrating x(1-x²) = x - x³) with no tricks or complications, making it easier than average for A-level statistics questions.
Spec	5.03c Calculate mean/variance: by integration

1 A random variable $X$ has probability density function f , where $$f ( x ) = \begin{cases} \frac { 3 } { 2 } \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find $\mathrm { E } ( X )$.

Show mark scheme Show mark scheme source

Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
$\frac{3}{2}\int_0^1(x-x^3)dx$	M1	Attempt to integrate $xf(x)$; ignore limits.
$=\frac{3}{2}\left[\frac{x^2}{2}-\frac{x^4}{4}\right]_0^1$	A1	Correct integration and limits.
$=\frac{3}{8}$	A1
3

## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{3}{2}\int_0^1(x-x^3)dx$ | **M1** | Attempt to integrate $xf(x)$; ignore limits. |
| $=\frac{3}{2}\left[\frac{x^2}{2}-\frac{x^4}{4}\right]_0^1$ | **A1** | Correct integration and limits. |
| $=\frac{3}{8}$ | **A1** | |
| | **3** | |

---

Show LaTeX source

1 A random variable $X$ has probability density function f , where

$$f ( x ) = \begin{cases} \frac { 3 } { 2 } \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

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

\hfill \mbox{\textit{CAIE S2 2023 Q1 [3]}}

This paper (8 questions)

View full paper

Q1 3 Q2 3 Q3 5 Q4 6 Q5 9 Q6 5 Q7 8 Q8 11