VCE Maths Methods Continuous Probability Mini Test 1
Number of marks: 10
Reading time: 2 minutes
Writing time: 15 minutes
Section A – Calculator Allowed
Instructions
• Answer all questions in pencil on your Multiple-Choice Answer Sheet.
• Choose the response that is correct for the question.
• A correct answer scores 1; an incorrect answer scores 0.
• Marks will not be deducted for incorrect answers.
• No marks will be given if more than one answer is completed for any question.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Let \( h \) be the probability density function for a continuous random variable \( X \), where
\[ h(x) = \begin{cases} \dfrac{x}{6} + k, & \text{for } -3 \leq x < 0 \\ -\dfrac{x}{2} + k, & \text{for } 0 \leq x \leq 1 \\ 0, & \text{elsewhere} \end{cases} \]
and \( k \) is a positive real number.
The value of \( \Pr(X < 0.5) \) is
- A. \(\frac{1}{2}\)
- B. \(\frac{15}{16}\)
- C. \(\frac{3}{16}\)
- D. \(\frac{49}{48}\)
A continuous random variable \( X \) has the following probability density function:
\[ g(x) = \begin{cases} \frac{x - 1}{20} & 1 \leq x < 6 \\ \frac{9 - x}{12} & 6 \leq x \leq 9 \\ 0 & \text{elsewhere} \end{cases} \]
The value of \( k \) such that \( \Pr(X < k) = 0.35 \) is
- A. \( \sqrt{14} - 1 \)
- B. \( \sqrt{14} + 1 \)
- C. \( \sqrt{15} - 1 \)
- D. \( \sqrt{15} + 1 \)
- E. \( 1 - \sqrt{15} \)
A continuous random variable, \( X \), has a probability density function given by
\[ f(x) = \begin{cases} \frac{2}{9}x e^{-\frac{1}{9}x^2}, & x \geq 0 \\ 0, & x < 0 \end{cases} \]
The expected value of \( X \), correct to three decimal places, is
- A. 1.000
- B. 2.659
- C. 3.730
- D. 6.341
- E. 9.000
The distribution of a continuous random variable, \(X\), is defined by the probability density function \(f\), where
\(f(x) = \begin{cases} p(x) & -a \le x \le b \\ 0 & \text{otherwise} \end{cases}\)
and \(a, b \in R^+\).
The graph of the function \(p\) is shown below.
It is known that the average value of \(p\) over the interval \([-a, b]\) is \(\frac{3}{4}\).
\(\Pr(X>0)\) is
- A. \(\frac{2}{3}\)
- B. \(\frac{3}{4}\)
- C. \(\frac{4}{5}\)
- D. \(\frac{7}{9}\)
- E. \(\frac{5}{6}\)
End of Section A
Section B – No Calculator
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Suppose that the queuing time, \( T \) (in minutes), at a customer service desk has a probability density function given by
\[ f(t) = \begin{cases} kt(16 - t^2) & 0 \leq t \leq 4 \\ 0 & \text{elsewhere} \end{cases} \] for some \( k \in \mathbb{R} \).
a. Show that \( k = \frac{1}{64} \). 1 mark
b. Find \( E(T) \). 2 marks
c. What is the probability that a person has to queue for more than two minutes, given that they have already queued for one minute? 3 marks
End of examination questions
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