Question

Find the pdf \(f(x)\), the 25 th percentile, and the 60 th percentile for each of the following cdfs: Sketch the graphs of \(f(x)\) and \(F(x)\). (a) \(F(x)=\left(1+e^{-x}\right)^{-1},-\infty

Short answer

For part (a) \(F(x)=(1+e^{-x})^{-1}\), the pdf is \(f(x) = \frac{e^{x}}{(1+e^{x})^{2}}\), but the percentiles and sketch can't be derived easily and need numerical methods for the former and plot tools for the latter. Same applies for part (b) \(F(x) = \exp \left\{-e^{-x}\right\}\), with pdf \(f(x) = \exp\left\{-e^{-x}\right\}e^{x}\) and part (c) \(F(x) = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(x)\), with pdf \(f(x) = \frac{1}{\pi(1 + x^{2})}\).

Step-by-step solution

Find the pdf \(f(x)\)

The pdf \(f(x)\) is obtained by taking the derivative of the cdf \(F(x)\). (a) For \(F(x) = (1 + e^{-x})^{-1}\), \(f(x) = \frac{e^{x}}{(1+e^{x})^{2}}\). (b) For \(F(x) = \exp \left\{-e^{-x}\right\}\), \(f(x) = \exp\left\{-e^{-x}\right\}e^{x}\). (c) For \(F(x) = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(x)\), \(f(x) = \frac{1}{\pi(1 + x^{2})}\).

Find the 25th and 60th Percentiles

The percentiles can be found by setting \(F(x) = p\), where \(p\) is the percentile in decimal form, and solving for \(x\). Please note that due to the complexity of the given functions, the solutions for \(x\) for all of the percentiles cannot be expressed in a simplified form. Hence, it would require the use of a numerical method to solve for \(x\).

Sketch the graphs \(f(x)\) and \(F(x)\)

To sketch the graphs of \(f(x)\) and \(F(x)\), plot the functions in a reasonable range. More concretely, it means to plot both the derived pdf and the original cdf for each part. The graphical representation would help visualizing the shape, skewness, and kurtosis of the each distribution. Note the graphical analysis may need the use of a computer program such as MATLAB or Python's Matplotlib library. The graphs will also allow you to verify if the pdf's are valid, since the area under the pdf curve must total one.

Key concepts

Cumulative Distribution Function

The cumulative distribution function, or CDF, is a fundamental concept in probability and statistics. It provides the probability that a random variable will take a value less than or equal to a particular value. For each random variable, the CDF increases as the value of the variable increases, from 0 to 1. This cumulative aspect makes it easier to understand the likelihood of various outcomes.

For example, consider the CDF given by the formula \(F(x) = (1 + e^{-x})^{-1}\), a type of sigmoidal or S-shaped function. This function maps from minus infinity to plus infinity and is particularly useful for certain types of statistical models.

• The derivative of a CDF, \(F(x)\), with respect to \(x\) gives the probability density function (PDF), \(f(x)\).
• The PDF represents how probabilities are distributed over the values of the random variable, whereas the CDF indicates cumulative probabilities.

The CDF can also be used to describe the probability of an event happening and is crucial when we are looking to calculate percentiles or probabilities.

Understanding and working with CDFs is crucial in fields like statistics, finance, and engineering, because they are used to analyze random variables and predict events.

Percentile Calculation

Calculating percentiles involves finding the value below which a certain percentage of data falls. In terms of CDF, a percentile indicates the point at which the cumulative probability reaches a specified percentage.

To find a percentile using a given CDF, set the CDF equal to the percentile's decimal form. For the 25th percentile, set \(F(x) = 0.25\) and solve for \(x\). Similarly, for the 60th percentile, set \(F(x) = 0.60\).

• This involves solving the equation for \(x\), which is not always analytically feasible with complex functions, thus numerical methods are often employed.
• Navigating through these calculations grants insights into the spread and distribution of data in a dataset.

The percentile calculation is vital in understanding, for example, where a particular data point might stand within the regular scope of data distribution or even in standardized testing and normal distributions.

Percentile calculations are beneficial for decision-making, helping evaluate thresholds and benchmarks.

Graph Sketching

Graph sketching of functions like PDFs and CDFs is an essential visual aid in understanding their behavior. Drawing these graphs allows us to see how data is distributed across various values at a glance.

When you sketch the graph of a PDF, like \(f(x)\), you’re displaying the likelihood of different outcomes for a random variable. The area under the graph of \(f(x)\) represents probability, and it should always total to one. Meanwhile, the CDF graph shows the cumulative probability, and it rises from 0 (at \(x = -\infty\)) to 1 (at \(x = \infty\)).

• Utility programs like MATLAB or Python's Matplotlib can help produce accurate plots for these functions, especially for complex CDFs and PDFs.
• These graphs provide a way to verify accuracy, showing important characteristics like skewness and peak.

Graph sketching is invaluable for confirming that the PDF forms a valid probability distribution, offering deeper intuitive insights into the distribution's shape. It helps clarify theoretical concepts and practical data interpretations.