Holooly Plus Logo
Holooly Plus Logo

Question 6.7: Suppose the time X (in hours) that an ATM outside a bank is ......

Suppose the time X (in hours) that an ATM outside a bank is in use during a day is a random variable with density function

f(x)= \begin{cases} \frac{x}{9}, \qquad \qquad \quad 0 \leq x \leq 3, \\ -\frac{1}{9}(x-6), \quad ~ 3 \lt x \leq 6, \end{cases}

Find the mean time that the machine operates during a day.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.
Report Answer

In this case, we have to split the range of integration in formula (6.8),

{E(X)=\int_{-\infty}^{\infty}x f(x){\mathrm{d}}x} \qquad \qquad (6.8)

because the expression for ƒ(x) given is different in the intervals [0, 3] and (3, 6]. Thus,

we obtain

\mu=E(X)=\int_{-\infty}^{\infty}x f(x)=\int_{0}^{3}{\frac{x}{9}}\;x\mathrm{d}x+\int_{3}^{6}x\left\{-{\frac{1}{9}}(x-6)\right\}\mathrm{d}x={\frac{1}{9}}\left[{\frac{x^{3}}{3}}\right]_{0}^{3}+\left(-{\frac{1}{9}}\right)\left\{\left[{\frac{x^{3}}{3}}\right]_{3}^{6}-6\left[{\frac{x^{2}}{2}}\right]_{3}^{6}\right\}=\frac{1}{9}\cdot\frac{3^{3}-0}{3}-\frac{1}{9}\left(\frac{6^{3}-3^{3}}{3}-6\frac{6^{2}-3^{2}}{2}\right)=3,

and so the machine operates on average 3 hours per day.

Related Answered Questions

Question: 6.2

The time, in minutes, between two consecutive arrivals of a train at a metro station is a continuous random variable X with distribution function F(t) = c − e^−t/3− t / 3 ⋅ e−t/3, t > 0, for some constant c. (i) Find the value of c and then obtain the density function f of X. (ii) Suppose that Mary ...

Verified Answer:

(i) In order to find the value of c, we use that t...
Question: 6.15

A random variable X has a mixed distribution whose discrete and continuous parts, ƒ1(x) and ƒ2(x), are described, respectively, by ƒ1(x) = {1/10, x = 4, 1/10, x = 8, and ƒ2(x) = {1/12, 0 < x < 4, 1/15, 8 < x ≤ 15. (i) What is the range of values for X? (ii) Calculate the probability P(2 < X < 9). ...

Verified Answer:

(i) The range for the discrete part of X is the se...
Question: 6.13

A nonnegative random variable X has distribution function F(x) = 1 – 5/3e^-5x – 2/5e^-2x, x ≥ 0. (i) Verify that F is a proper distribution function. (ii) Find E(X). ...

Verified Answer:

(i) We need to check that all conditions for a dis...
Question: 6.12

Prove that there does not exist a random variable X with E(X) = 𝜇 and Var(X) = 𝜎² such that P(𝜇 − 2𝜎 < X < 𝜇 + 2𝜎) = 0.70. ...

Verified Answer:

From Chebyshev’s inequality we have, for any varia...
Question: 6.8

At one end of a stick with length OA = R, we place a sphere. We push the sphere using a force of random magnitude so that it is rotated around point O (see Figure 6.6) creating an angle of Θ degrees, and the density function of this angle is given by ƒΘ(θ) = { 2/π, 0 < θ < π/2, 0, elsewhere. Find ...

Verified Answer:

As can be seen from Figure 6.6, we have X = R cos ...
Question: 6.5

(Measurement error) It has been found that an electronic device exhibits a measurement error, denoted by X, which is regarded as a random variable with density function f(x) = {1/7, 3 ≤ x ≤ c, 0, elsewhere, for some suitable constant c > 0, which represents the maximum value of that error. (i) Find ...

Verified Answer:

(i) Since we are given that ƒ is a probability den...
Question: 6.4

Suppose that some amount of oil is spilled onto the surface of an ocean (for instance, due to an oil-tanker accident). The oil spot then spreads away on the sea surface. The radius, R, of the oil spot in kilometers, 24 hours after the accident, has density function ...

Verified Answer:

Let Y be the random variable denoting the size of ...
Question: 6.3

(Distribution for a linear transformation of a random variable) Let X be a continuous random variable with density function ƒ and let Y = aX + b, be another random variable that arises as a linear transformation of X; here, a > 0 and b are given real numbers. (i) Find an expression for the density ...

Verified Answer:

Let F and F_Y be the distribution f...
Question: 6.1

The time required, in hours, to repair a car at a garage is a random variable X with density function f(x) = c(4x − x²), 0 < x ≤ 4. (i) Find the value of the constant c. (ii) Find the probability that for a car which arrives now at the garage, the amount of time needed to get repaired will be (a) ...

Verified Answer:

(i) For ƒ given in the example to be a probability...