Question

Let \(f(x)=1 / x^{2}, 1

Short answer

The probability \(P_{X}\left(C_{1} \cup C_{2}\right) = P_{X}(C_{1}) + P_{X}(C_{2})\) and the probability \(P_{X}\left(C_{1} \cap C_{2}\right) = 0\).

Step-by-step solution

Calculate the Probability for Set \(C_{1}\)

Calculate the integral of the given pdf \(f(x)\) over the set \(C_{1} = \{x : 1 < x < 2\}\). This is done by integrating the function \(\int_{1}^{2} f(x) dx = \int_{1}^{2} 1/x^{2} dx\).

Calculate the Probability for Set \(C_{2}\)

Repeat the same procedure for set \(C_{2} = \{x : 4 < x < 5\}\). The computation is \(\int_{4}^{5} f(x) dx = \int_{4}^{5} 1/x^{2} dx\).

Calculate the Probability for Union of Sets

The union of sets \(C_{1}\) and \(C_{2}\) is the set of elements which are in \(C_{1}\), in \(C_{2}\), or in both. Since \(C_{1}\) and \(C_{2}\) do not intersect, their union is equal to the sum of their individual probabilities. Hence, \(P_{X}\left(C_{1} \cup C_{2}\right) = P_{X}(C_{1}) + P_{X}(C_{2})\).

Calculate the Probability for Intersection of Sets

The intersection of sets \(C_{1}\) and \(C_{2}\) is the empty set (since they do not overlap). Thus, the probability of their intersection, \(P_{X}\left(C_{1} \cap C_{2}\right)\), is 0. Hence, the probability is 0.

Key concepts

Mathematical Statistics

Under the umbrella of mathematical statistics lies a crucial concept known as the probability density function (pdf), a centerpiece for understanding how probabilities are distributed across continuous random variables. In simple terms, a probability density function quantifies the likelihood of a random variable falling within a certain range. When provided with a pdf such as \(f(x) = 1 / x^2\), defining the probability of finding the random variable \(X\) within specified intervals becomes straightforward through mathematical statistics tools.

For example, to find the probability that \(X\) lands in the interval \(C_1\) or \(C_2\), we calculate the integral of the pdf over these intervals. This mirrors how we might measure an area under a curve in basic calculus to represent the probability over that interval. It is important to understand that the integral calculation is not just a mechanical step but reflects the total 'weight' of probability within the specified range.

Integral Calculus

Integral calculus is a branch of mathematics focused on the concepts of integration and the finding of functions' antiderivatives. In the context of probability and statistics, integral calculus is used to compute the area under a curve of a pdf, which corresponds to the probability of the random variable falling within a specific range.

This application is seen in our exercise, where an integral calculates the probabilities for the sets \(C_1\) and \(C_2\) by integrating the given pdf over these intervals. The use of an integral from \(1\) to \(2\) to find \(P_X(C_1)\) and from \(4\) to \(5\) to find \(P_X(C_2)\) demonstrates the practicality of integral calculus in solving real-world problems. This process is akin to summing up an infinite series of infinitesimally small rectangles under the curve between the limits of integration, thereby providing the enclosed 'area' which represents our desired probability.

Probability Theory

Probability theory provides a framework for quantifying the uncertainty of events and is foundational to the field of statistics. It encompasses important principles such as the addition rule, which states that the probability of the union of two events equals the sum of the probabilities of each event happening separately, provided the events are mutually exclusive – meaning they cannot both occur at the same time.

In our example, the probability of the union of sets \(C_1\) and \(C_2\), denoted as \(P_X(C_1 \cup C_2)\), is simply the sum of their individual probabilities because the sets do not intersect, reflecting their mutual exclusivity. Moreover, the probability of the intersection of sets \(C_1\) and \(C_2\), \(P_X(C_1 \cap C_2)\), is zero, as they cannot happen simultaneously – they have no elements in common. The application of these probability theory rules simplifies the computation process and helps to illuminate the relationship between the probabilities of complex events.