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Exam-Style Questions on Probability Density Function

Problems on Probability Density Function adapted from questions set in previous Mathematics exams.

1.

IB Analysis and Approaches

A continuous random variable \(X\) has probability density function \(f\) defined by

$$ f(x) = \begin{cases} \dfrac{1}{3q}, & 2q \le x \le 5q \\ 0, & \text{ otherwise } \end{cases}$$

where \(q\) is a positive real number.

(a) State \(E(X)\) in terms of \(q\).

(b) Use integration to find \(Var(x)\) in terms of \(q\).


2.

IB Analysis and Approaches

The continuous random vanable \(X\) has probability density function:

$$ f(x) = \begin{cases} \dfrac{k}{\sqrt{16-x^2}}, & 0 \le x \le 2 \\ 0, & \text{ otherwise } \end{cases}$$

(a) Find the value of \(k\)

(b) Show that \(E(X) = \dfrac{12(2-\sqrt{3})}{\pi}\)


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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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