Question

For each of the following cdfs \(F(x)\), find the pdf \(f(x)[\mathrm{pmf}\) in part \((\mathbf{d})]\), the first quartile, and the \(0.60\) quantile. Also, sketch the graphs of \(f(x)\) and \(F(x)\). May use \(\mathrm{R}\) to obtain the graphs. For Part(a) the code is provided. (a) \(F(x)=\frac{1}{2}+\frac{1}{\pi} \tan ^{-1}(x),-\infty

Short answer

The main steps in solving this problem involve calculus (for finding the derivative of the CDF to get the PDF) and algebra (for solving for the first quartile and 0.60 quantile). After these are determined, the graphing of \(f(x)\) and \(F(x)\) can be achieved using software or manual drawing depending on the complexity of the functions. The results will vary for each function from (a) to (d).

Step-by-step solution

Find the PDF

To find the PDF for each CDF, we need to take the derivative of \(F(x)\) with respect to \(x\). For (a): The derivative of \(F(x)= \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(x)\) is \(f(x) = \frac{1}{\pi(1+x^2)}\). Similarly, for (b), (c) and (d), derivatives can be taken to get the respective PDFs.

Determine the first quartile and 0.60 quantile

The first quartile is the value of \(x\) such that \(F(x) = 0.25\) and the 0.60 quantile is the value of \(x\) such that \(F(x) = 0.60\). To find these values, set \(F(x) = 0.25\) and \(F(x) = 0.60\) respectively and solve for \(x\). This might involve numerical methods depending on the complexity of \(F(x)\).

Sketch the Graphs of \(f(x)\) and \(F(x)\)

The graphs of \(f(x)\) and \(F(x)\) can be drawn using software like R or done manually if the functions are simple. The function \(F(x)\) should be a non-decreasing function, while \(f(x)\) as the derivative of \(F(x)\) denotes the rate of change of \(F(x)\) and will be centered around the mode. It's essential to make sure that these properties hold for each graph.

Key concepts

Probability Density Function (PDF)

The probability density function (PDF) is a mathematical expression that describes the likelihood of a continuous random variable taking on a specific value. In essence, a PDF gives us an understanding of the distribution of probabilities across the range of possible outcomes. For instance, the PDF derived from the cumulative distribution function (CDF) by differentiating it with respect to the variable. In the exercise, the PDF for the given CDF in part (a) is \(f(x) = \frac{1}{\pi(1+x^2)}\), which indicates the probability distribution for a continuous variable over its possible range.

Understanding the PDF is crucial in statistical analysis as it helps in finding probabilities for intervals rather than specific points. To calculate the probability that a random variable falls within a certain range, one would integrate the PDF over that range.

Quantiles

Quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities. Common quantiles include the quartiles, which divide the data into quarters, and the percentiles, which divide the data into hundred equal parts. In statistical terms, quantiles represent the points at which cumulative probability reaches a certain threshold.

In the given exercise, the first quartile is at the point where the CDF is 0.25, signaling that 25% of the distribution lies below this value. Similarly, the 0.60 quantile is where the CDF is 0.60, meaning 60% of the values lie below and 40% lie above this point. Identifying quantiles is valuable for summarizing data and understanding its distribution, especially in contexts like performance evaluations or analyzing income distributions.

Mathematical Statistics

Mathematical statistics involves the study of statistics using the principles and methods of mathematics. It includes the theoretical underpinnings of statistical measures, tests, and analyses. The field encompasses the derivation of statistical methods, the establishment of statistical theorems, and the development of computational techniques for performing statistical operations.

At its core, mathematical statistics deals with creating models to represent complex data and making inferences about populations from samples. The exercise provided incorporates various elements of mathematical statistics, such as determining PDFs from CDFs and computing quantiles, which are foundational concepts in analyzing and interpreting data.

Numerical Methods

In situations where analytical solutions are difficult or impossible to obtain, numerical methods come to the rescue. Numerical methods are algorithms used for approximating mathematical operations and are essential for handling complex equations in statistics. For instance, when finding quantiles requires solving equations that don't have simple algebraic solutions, numerical algorithms such as Newton-Raphson, bisection method, or Monte Carlo simulations can be used.

These techniques are indispensable for modern statistical analysis. In the exercise, numerical methods may be necessary to accurately compute the first quartile and 0.60 quantile for some of the CDFs, especially when they involve transcendental equations that cannot be solved explicitly.

Graphing Distributions

The visual representation of distributions is a potent way to understand and communicate statistical data. Graphing distributions allows us to see the shape, spread, and central tendencies of the data. For continuous random variables, graphing both the CDF and PDF provides insights into their behavior and properties. The CDF graph is always non-decreasing, approaching 1 as the variable gets larger, while the PDF graph shows the densities that correspond to specific points along the distribution.

The graphical output also significantly aids in identifying outliers, patterns, or anomalies in data sets. In the educational context of the given exercise, sketching graphs serves not only as an analytical tool but also as a learning aid that helps to visually grasp the concepts of CDF and PDF.

R Programming for Statistics

R is a widely used programming language and software environment in the field of statistics for data analysis and graphical representation. Its comprehensive statistical and graphical capabilities make it an excellent tool for carrying out intricate statistical computations and plotting distributions.

In the task provided, R could be utilized to derive and graph the CDF and PDF of each part, thereby providing a visual understanding of the solution. Using R's functions and libraries can simplify the process of solving statistical problems and validating solutions. By writing appropriate code in R, one can easily compute the PDFs, quantiles, and create detailed graphs of the distributions as required by the exercise.