Question

The median and quantiles, in general, are discussed in Section 1.7.1. Find the median of each of the following distributions: (a) \(p(x)=\frac{4 !}{x !(4-x) !}\left(\frac{1}{4}\right)^{x}\left(\frac{3}{4}\right)^{4-x}, x=0,1,2,3,4\), zero elsewhere. (b) \(f(x)=3 x^{2}, 0

Short answer

The median for the first function can be found by sequentially adding the probabilities until reaching or exceeding 0.5. The median of the second and third functions would require calculating the cumulative distribution function, setting it equal to 0.5, and solving for x. The third function might require numerical methods due to its complexity.

Step-by-step solution

Find the Median for the First Distribution

Given the discrete probability distribution function \(p(x)=\frac{4 !}{x!(4-x)!}\left(\frac{1}{4}\right)^x\left(\frac{3}{4}\right)^{4-x}, x=0,1,2,3,4\), zero elsewhere. Start from x=0 and keep adding the probability \(p(x)\) until the cumulative probability is greater than or equal to 0.5.

Find the Median for the Second Distribution

Given the continuous probability distribution function \(f(x)=3x^2, 0<x<1\), zero elsewhere. The cumulative distribution function \(F(x)\) is calculated by taking the integral of \(f(x)\) from 0 to x. The median is the value of x for which \(F(x)=0.5\). Thus, calculate the integral of \(f(x)\), set it equal to 0.5, and solve for x.

Find the Median for the Third Distribution

Given the continuous probability distribution function \(f(x)=\frac{1}{\pi(1+x^2)}, -∞<x<∞\). The cumulative distribution function \(F(x)\) is calculated by taking the integral of \(f(x)\) from -∞ to x. The median is the value of x for which \(F(x)=0.5\). Because of the complexity of this function, the integral and solution may need to be obtained using numerical methods.

Key concepts

Probability Distribution Function

Understanding the probability distribution function (PDF) is fundamental when dealing with statistical distributions. A PDF, denoted as either p(x) for discrete random variables or f(x) for continuous random variables, describes the likelihood of each outcome within a set of possibilities. For discrete distributions, like the one in exercise part (a), where the function is given by p(x) = \( \frac{4!}{x!(4-x)!}\left(\frac{1}{4}\right)^x\left(\frac{3}{4}\right)^{4-x}, x=0,1,2,3,4\), the PDF tells us the probability of each integer value of x.

For continuous distributions, like in parts (b) and (c) with their respective functions, the PDF describes how probability density is distributed over intervals of values rather than specific points. For example, in part (b), f(x) = 3x^2, 0 < x < 1, the density of the probabilities is illustrated by the shape of the curve over the interval of x. The integral of a continuous PDF over its entire range will always equal 1, representing the certainty that some outcome within the range will occur.

Cumulative Distribution Function

The cumulative distribution function (CDF), usually denoted by F(x), is closely linked to the PDF and provides a cumulative probability associated with a value. It tells us the probability that a random variable will take on a value less than or equal to x. For discrete random variables, the CDF is the sum of the probabilities of the outcomes up to and including x, as demonstrated in Step 1 of the exercise's solution. In this step-by-step calculation, we begin with x = 0 and continue to add each subsequent probability until the sum reaches or exceeds 0.5, which indicates the median.

For continuous variables, calculating the CDF involves integrating the PDF from a lower bound up to x. As seen with parts (b) and (c) in Step 2 and Step 3, we are looking for the value of x where F(x) = 0.5. This process effectively tells us the median of the distribution, which is the point where half the probability weight of the distribution lies to the left and half to the right. This task can be straightforward for functions with simple integral forms but may require more complex numerical methods for integrals that do not have closed-form solutions.

Quantiles

Quantiles are points in a distribution that divide the data into intervals of equal probability. The median, which is the focus in our exercise, is a special type of quantile that divides data into two equal halves; in other words, it is the 0.5 or 50th percentile. Other common quantiles include quartiles, which divide the data into four equal parts, and percentiles, which divide the data into 100 equal parts.

When calculating quantiles for a discrete distribution, one may need to consider the exact cumulative probabilities (as in part (a) of the exercise) to find the smallest value of x where the CDF is greater than or equal to the desired percentile. For continuous distributions, the approach involves solving for x in the equation F(x) = p, where p is the percentile in decimal form. In our step-by-step solutions for parts (b) and (c), finding the median requires setting p to 0.5 and solving for x in the CDF. This can generally be accomplished through algebraic methods or numerical approximation, depending on the complexity of the PDF and its integral.