Question

Consider the cdf \(F(x)=1-e^{-x}-x e^{-x}, 0 \leq x<\infty\), zero elsewhere. Find the pdf, the mode, and the median (by numerical methods) of this distribution.

Short answer

The probability density function, mode, and median of the specified distribution can be found by following this approach: first differentiate the cdf to find the pdf, next find the derivative of the pdf and solve for x to find the mode, and finally, set the cdf equal to 0.5 and solve for x to find the median using numerical methods.

Step-by-step solution

Find the Probability Density Function (pdf)

The pdf is the derivative of the cdf. This can be calculated by differentiating the given function \(F(x)=1-e^{-x}-x e^{-x}\) w.r.t. \(x\). Hence, \(f(x) = \frac{dF(x)}{dx}\).

Find the mode of the distribution

The mode is the value that maximizes the pdf. In other words, it is the solution for \(x\) to the equation \(f'(x) = 0\), where \(f'(x)\) is the first derivative of the pdf function. After finding this derivative, set it equal to zero and solve for \(x\).

Find the median of the distribution numerically

The median of a probability distribution is the value for which the cumulative distribution function equals 0.5. Form an equation setting \(F(x) = 0.5\) and solve for \(x\) numerically using a root-finding algorithm.

Key concepts

Cumulative Distribution Function

Understanding the cumulative distribution function (CDF) is crucial in statistics. Think of the CDF as a way to accumulate probabilities up to a certain point. It gives the probability that a random variable is less than or equal to a certain value. In mathematical terms, for a continuous random variable, the CDF at a point x is defined as the integral of the probability density function (pdf) from negative infinity to x.

For the example given in the exercise, the CDF, denoted as F(x), is specified for values of x greater than or equal to zero and less than infinity. It signifies that we're dealing with a positive continuous random variable. To find the pdf, which is the derivative of the CDF, you simply apply calculus to differentiate F(x) with respect to x. This transformation from CDF to pdf is essential when trying to find other distribution characteristics, such as the mode or the median.

Mode of a Distribution

The mode of a distribution represents the value that appears most frequently in a data set. In a continuous probability distribution, it's the peak of the probability density function. To locate the mode, we look for the maximum point on the pdf curve, which is where its first derivative with respect to x is zero.

Referring to the given exercise, finding the mode involves taking the derivative of the pdf, denoted f(x), and solving the equation where this derivative is equal to zero. In some cases, there may be more than one mode, which indicates we have a multimodal distribution. However, for the majority of textbook cases, we focus on finding a single, most prominent peak, or mode, of the distribution.

Median of a Distribution

The median of a distribution is the value that divides the probability in half. It is the point at which half of the area under the pdf is to the left, and the other half to the right. Formally, it's the value m such that the CDF F(m) equals 0.5. This concept is particularly helpful in understanding the central tendency of a distribution that isn't skewed by extreme values like outliers.

For the exercise in question, we find the median by setting up the equation F(x) = 0.5 and solving for x. Numerical methods come into play here, as closed-form solutions might not be possible or easy to determine. This application of numerical methods to find the median showcases the practical approach statisticians must sometimes take when dealing with complex distributions.

Numerical Methods in Statistics

Numerical methods are algorithms used to approximate numerical solutions to statistical problems. These methods are indispensable when dealing with equations that are too complex to solve analytically. Common numerical techniques include root-finding algorithms such as the Newton-Raphson method or the bisection method, numerical integration, and optimization algorithms.

In the given exercise, a numerical method would be used to solve the equation F(x) = 0.5 for the median, since finding an exact analytical solution for x might not be straightforward. These methods allow for an iterative approach, where an initial guess is refined multiple times until a sufficiently accurate solution is found. This highlights the blend of theoretical knowledge and computational skills required in modern statistics.