Continuous random variables

The number of minutes, T, by which a morning train arrives late is modelled by the continuous random variable with probability density function f(t) = kt² for 0 ≤ t ≤ 3, and f(t) = 0 otherwise, where k is a constant.

The length, X metres, of the offcut left when a roll of cable is used is modelled by a continuous random variable with cumulative distribution function F(x) = 0 for x < 0, F(x) = (x² + 2x)/24 for 0 ≤ x ≤ 4, and F(x) = 1 for x > 4.

The time, X hours, that a student takes to complete a logic puzzle has probability density function f(x) = 3x²/64 for 0 ≤ x ≤ 4, and f(x) = 0 otherwise.

The distance, X km, that a delivery drone travels from its depot before its first drop-off has probability density function f(x) = (4 − x)/8 for 0 ≤ x ≤ 4, and f(x) = 0 otherwise.

The time, X minutes, that a visitor spends at an interactive museum display is modelled by a continuous random variable. Figure 1 shows the graph of the probability density function f of X. The graph is a triangle with vertices at the origin, (2, k) and (5, 0), and f(x) = 0 outside 0 ≤ x ≤ 5.

Breakdowns on a 6 km stretch of road are recorded. The distance, X km, from the start of the stretch to the point of a breakdown is modelled by a continuous random variable. Figure 1 shows the graph of the cumulative distribution function F of X. It consists of straight line segments joining (0, 0), (2, 0.2), (4, 0.8) and (6, 1), with F(x) = 0 for x < 0 and F(x) = 1 for x > 6.

The distance, X cm, from the centre of a dartboard to the point where a dart lands is modelled by the probability density function f(x) = k(4 − x²) for 0 ≤ x ≤ 2, and f(x) = 0 otherwise, where k is a constant. Figure 1 shows the graph of f. It meets the vertical axis at the point (0, 0.75).

The time, X minutes, that a passenger waits for a lift is modelled by a continuous random variable whose probability density function is shown in Figure 1. The density is constant on each of the intervals 0 ≤ x < 1, 1 ≤ x < 2 and 2 ≤ x ≤ 4, with heights 0.25, h and 0.125 respectively, and is zero outside 0 ≤ x ≤ 4.

The waiting time, X minutes, of a customer at a pharmacy counter is modelled by the probability density function f(x) = kx for 0 ≤ x ≤ 2, f(x) = 2k for 2 < x ≤ 5, and f(x) = 0 otherwise, where k is a constant.

The daily rainfall, X cm, at a weather station during the wet season is modelled by the probability density function f(x) = x/5 for 0 ≤ x ≤ 2, f(x) = 2/5 for 2 < x ≤ 3.5, and f(x) = 0 otherwise.

The time, X minutes, for a chemical indicator to change colour is modelled by the probability density function f(x) = k(x − 1)(5 − x) for 1 ≤ x ≤ 5, and f(x) = 0 otherwise, where k is a constant.

The breaking strength, X kN, of a length of thin nylon cord is modelled by the probability density function f(x) = kx(9 − x²) for 0 ≤ x ≤ 3, and f(x) = 0 otherwise, where k is a constant.

Seeds from a particular plant have diameters between 2 mm and 4 mm. For a seed chosen at random, the probability that its diameter X mm is at most x is (x² − 4)/12, for 2 ≤ x ≤ 4. This gives the cumulative distribution function F(x) of X on that interval.

A machine cuts square sheets of card. The side length of a sheet, X decimetres, is modelled by the probability density function f(x) = x/4 for 1 ≤ x ≤ 3, and f(x) = 0 otherwise. The area of a sheet is A = X² square decimetres.

The distance, X metres, from one end of a 5 m beam to the point where a crack first appears is modelled by a continuous random variable. Figure 1 shows the graph of its probability density function: a straight line from (0, 0) to (1, h), a horizontal line at height h from x = 1 to x = 4, and a straight line from (4, h) to (5, 0). The density is zero outside 0 ≤ x ≤ 5.

A bus shelter has a bench 4 m long. The distance, X metres, from the left-hand end of the bench to the point where the first passenger sits is modelled by the probability density function f(x) = k(x − 2)² for 0 ≤ x ≤ 4, and f(x) = 0 otherwise, where k is a constant. Figure 1 shows the graph of f.

The time, X minutes, that a caller waits to be answered by a helpline is modelled by the probability density function f(x) = 2(3 − x)/9 for 0 ≤ x ≤ 3, and f(x) = 0 otherwise. To test the model, the waiting times of 60 callers were recorded. Figure 1 is a box plot of these 60 times; the vertical gridlines are 0.1 minutes apart.

A drill makes a hole at a random position along a metal strip 4 cm long. The distance, X cm, of the hole from the left-hand end of the strip has the probability density function shown in Figure 1. The graph consists of two straight line segments, joining (0, 0.1) to (2, 0.4) and (2, 0.4) to (4, 0.1), and f(x) = 0 outside 0 ≤ x ≤ 4.

In a game, a cork is thrown as far as possible beyond a line. The distance, X metres, beyond the line at which it lands is modelled by the probability density function f(x) = k(1 + x)² for 0 ≤ x ≤ 2, and f(x) = 0 otherwise, where k is a constant. Figure 1 shows the graph of f. The shaded region, between x = 0 and x = a, has area 0.25.

The drying time, T minutes, of a sample of a new paint is modelled by the probability density function f(t) = kt(4 − t) for 0 ≤ t ≤ 4, and f(t) = 0 otherwise, where k is a constant. To test the model, the drying times of 200 samples were recorded. Figure 1 is a histogram of these times.

The thickness, X mm, of a protective coating sprayed onto a panel is modelled by the probability density function f(x) = a + bx for 1 ≤ x ≤ 3, and f(x) = 0 otherwise, where a and b are constants. Given that E(X) = 2.1,

The time, X hours, for which a machine component works before it needs adjusting is modelled by a continuous random variable with cumulative distribution function F(x) = 0 for x < 0, F(x) = kx² for 0 ≤ x ≤ 2, F(x) = 1 − c(5 − x)² for 2 < x ≤ 5, and F(x) = 1 for x > 5, where k and c are constants. Given that P(X ≤ 2) = 0.4,

The time, X hours, taken by an engineer to repair a fault is modelled by the probability density function f(x) = k(3 − x)² for 0 ≤ x ≤ 3, and f(x) = 0 otherwise, where k is a constant. The cost of a repair, C pounds, is C = 40 + 25X.

The distance, X metres, that a ball rolls after leaving a ramp is modelled by the probability density function f(x) = kx² for 0 ≤ x ≤ 2, f(x) = k(6 − x) for 2 < x ≤ 6, and f(x) = 0 otherwise, where k is a constant.

The time, X hours, for which a hired e-scooter is used is modelled by a continuous random variable with probability density function f, shown in Figure 1. For 0 ≤ x ≤ 2, f(x) = k(4x − x²), where k is a positive constant. For 2 < x ≤ 4, the graph of f is the straight line segment from P(2, 4k) to (4, 0). Also f(x) = 0 outside 0 ≤ x ≤ 4.

The time, X hours, for which a kayak hired from a lake centre is out on the water is at most 4 hours. X has cumulative distribution function F(x) = 0 for x < 0, F(x) = ax + bx² for 0 ≤ x ≤ 4 and F(x) = 1 for x > 4, where a and b are constants. Figure 1 shows the graph of F, which passes through the points (2, 0.4) and (4, 1).

A rod AB of length 6 cm is cut at a point P, where AP = X cm. The two pieces, of lengths X cm and (6 − X) cm, are used as the sides of a rectangle, as shown in the upper part of Figure 1. The random variable X has the probability density function shown in the lower part of Figure 1: a triangle with vertices at the origin, (3, h) and (6, 0), with f(x) = 0 outside 0 ≤ x ≤ 6. The area of the rectangle is A cm².

The time, X hours, that a cyclist on a long-distance ride spends at a rest stop is modelled by the probability density function f(x) = kx²(a − x) for 0 ≤ x ≤ a, and f(x) = 0 otherwise, where k and a are positive constants. Figure 1 shows the graph of f, which has a maximum point at M.