Understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts.

The time in minutes between the arrival of buses at a particular stop is denoted by the continuous random variable X. The cumulative distribution function of X is defined by F(x) = { 1 - 16/x², 4 ≤ x ≤ 8 { 0, x < 4 { 1, x > 8 (a) Determine the probability density function f(x) of X. [2 marks] (b) Determine the probability that the time between consecutive buses is between 5 and 6 minutes. [2 marks] (c) Determine the expected value of X, given that E(X) = ∫₄⁸ x f(x) dx. [1 mark]

The continuous random variable X has the probability density function f(x) = { 2x, 0 ≤ x ≤ 1 { 0, otherwise Determine P(0.25 ≤ X ≤ 0.75).

A continuous random variable $X$ has the probability density function $$f(x) = \begin{cases} \frac{2}{5}(1 + x) & \text{for } 0 \le x \le 1 \\ 0 & \text{otherwise} \end{cases}$$ Determine $P(0 \le X \le 0.5)$.

The diameter, in millimetres, of ball bearings manufactured by a machine has a cumulative distribution function \( F(d) \) defined by \[ F(d) = \begin{cases} 0, & d < 15 \\ \frac{(d - 15)^3}{125}, & 15 \leq d \leq 20 \\ 1, & d > 20 \end{cases} \] where \( d \) represents the diameter in millimetres. Determine the probability that a randomly selected ball bearing has a diameter that meets the premium grade specification given in the stimulus table.

A manufacturing plant monitors the temperature fluctuations, \(T\) degrees Celsius, in a critical production chamber. The temperature variation from the target temperature can be modelled by a continuous random variable \(X\), where \(T = X + 85\). The probability density function of \(X\) is given by \[ f(x) = \begin{cases} k(9 - x^2), & -3 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases} \] where \(k\) is a positive constant. The table below shows the cumulative distribution function \(F(x)\) evaluated at selected values. (a) Determine the value of \(k\). (2 marks) (b) Determine the probability that the actual temperature in the chamber is between 83°C and 86°C. (2 marks) (c) Use the information in the table to determine \(P(X \leq 1.5)\). (1 mark)