Question

Let \(x\) denote the amount of gravel sold (in tons) during a randomly selected week at a particular sales facility. Suppose that the density curve has height \(f(x)\) above the value \(x\), where $$ f(x)=\left\\{\begin{array}{ll} 2(1-x) & 0 \leq x \leq 1 \\ 0 & \text { otherwise } \end{array}\right. $$ The density curve (the graph of \(f(x)\) ) is shown in the following figure: Use the fact that the area of a triangle \(=\frac{1}{2}\) (base)(height) to calculate each of the following probabilities: a. \(P\left(x<\frac{1}{2}\right)\) b. \(P\left(x \leq \frac{1}{2}\right)\) c. \(P\left(x<\frac{1}{4}\right)\) d. \(P\left(\frac{1}{4}

Short answer

The probabilities asked for in the problem are: a. 1/8, b. 1/8, c. 3/32, d. 1/32, e. 7/8, f. 29/32.

Step-by-step solution

Calculate P(x < 1/2)

The probability \(P(x < 1/2)\) is the area of a triangle with base \(x = 1/2\) and height \(f(x) = 2(1 - 1/2) = 1\). Hence, \(P(x < 1/2) = (1/2) * 1/2 * 1 = 1/8\).

Calculate P(x

The probability \(P(x <= 1/2)\) is equal to the probability \(P(x < 1/2)\) because for continuous random variables, the probabilities at individual points are zero. Thus, \(P(x <= 1/2) = 1/8\).

Calculate P(x < 1/4)

The probability \(P(x < 1/4)\) is the area of a triangle with base \(x = 1/4\) and height \(f(x) = 2(1 - 1/4) = 3/2\). Hence, \(P(x < 1/4) = (1/2) * 1/4 * 3/2 = 3/32\).

Calculate P(1/4 < x < 1/2)

The probability \(P(1/4 < x < 1/2)\) is the difference between the areas of two triangles: one with base \(x = 1/2\) (calculated in Step 1), and another with base \(x = 1/4\) (calculated in Step 3). Thus, \(P(1/4 < x < 1/2) = P(x < 1/2) - P(x < 1/4) = 1/8 - 3/32 = 1/32\).

Calculate the probability that gravel sold exceeds 1/2 ton

The probability that gravel sold exceeds 1/2 ton is equal to 1 minus the probability that it is less than or equal to 1/2 ton. Thus, \(P(x > 1/2) = 1 - P(x <= 1/2) = 1 - 1/8 = 7/8\).

Calculate the probability that gravel sold is at least 1/4 ton

The probability that gravel sold is at least 1/4 ton is equal to 1 minus the probability that it is less than 1/4 ton. Thus, \(P(x >= 1/4) = 1 - P(x < 1/4) = 1 - 3/32 = 29/32\).

Key concepts

Continuous Random Variables

When we talk about continuous random variables, we refer to quantities that can take an infinite number of possible values within a given range. For example, the amount of gravel sold, measured in tons, is a continuous random variable because it can potentially be any real number within the possible selling capacity. Unlike discrete random variables, which have countable values (like the number of textbooks on a shelf), continuous random variables require an associated probability density function (PDF) to describe the likelihood of the variable being in a specific range. The key feature distinguishing continuous variables is that the probability of it assuming any single, exact value is zero; instead, we focus on intervals to determine probabilities.

For instance, if we consider the exercise at hand, the random variable 'x,' representing gravel sold, is treated as a continuous random variable since x can take on any value within the range (0, 1). Probabilities are computed using the PDF, not by counting individual outcomes as you would with discrete variables.

Area under the Curve

The probability density function (PDF) for a continuous random variable is graphically represented by a curve on a graph. The probability of the variable falling within a certain range is determined by the area under the curve between two points. Because of this, understanding the geometry behind the curve becomes necessary. The total area under the curve for a properly defined PDF always equals 1, signifying a 100% probability that the variable will take on a value within the given range.

In the exercise, the PDF of the random variable 'x' is defined by the function \(f(x)\) and considering the curve is a straight line forming a triangle in this case, calculating areas is straightforward using the familiar formula for the area of a triangle. It's important to realize that calculating probabilities involves finding the area of various shapes under the curve, where the x-axis represents the possible values and the y-axis shows the density.

Probability Calculations

In probability theory, especially for continuous random variables, it is often necessary to calculate the likelihood, or probability, that the variable falls within a certain range. These calculations are derived from the properties of the PDF and the corresponding graphical area under the curve. To execute these calculations correctly, one must analyze the shape of the graph and use the appropriate geometric formulas.

Referring back to our gravel example 'x', probabilities such as \( P(x < 1/2) \) or \( P(x > 1/2) \) are deduced by considering the shape under the curve—triangles in this case—and applying the area formula. Thus, taking the appropriate base and height from the function \(f(x)\) and implementing the formula for the area of a triangle provides the solution to various probability inquiries detailed in the exercise. Learning to visualize and calculate the area is a critical skill for effective probability analysis with continuous random variables.