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 pdf is the derivative of the given \(F(x)\). The mode is the x-value where the derivative of the pdf equals zero. And the median is calculated by setting \(F(x)=0.5\) and solving the obtained equation by numerical methods. Exact answer values depends on the computational results from the numerical method used.

Step-by-step solution

Find the PDF

Recall that the pdf is derived from the cdf by taking its derivative. So, we'll differentiate \(F(x)=1-e^{-x}-x e^{-x}\) with respect to \(x\). Using the chain rule of differentiation, we'll obtain the derivative for each term.

Solve for the Mode

In order to find the mode of this distribution, we need to find the maximum value of the pdf. This can be done by setting the derivative of the pdf to zero and solve for x. For the equation obtained in step 1, make derivative equals zero and solve for x to find the mode.

Compute for Median

The median is the value of \(x\) for which the cumulative distribution function equals 0.5. In this case, \(F(x) = 0.5\), we solve this equation for \(x\) using numerical methods like bisection method or Newton-Raphson method. This needs programming/software skills to solve such equations numerically. Use a software tool or a calculator that can handle numerical computations.