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Exam-Style Questions on Probability Density FunctionProblems 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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