Solved – How to calculate the median of a pdf

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I'm new to statistics and I'm struggling to solve a question from an assignment. I have a probability density function and I need to calculate its median

Here is the function:

$$f(x) = 2xe^{-x^2}, x>= 0$$

The answer is $\sqrt{\log 2}$.

Can someone help? I'm trying to learn

Best Answer

A median by definition is a real number $m$ that satisfies $$P(X\leq m)=\frac{1}{2}.$$ So in your case, we have $$\int_0^m2xe^{-x^2}\,dx=\frac{1}{2}.$$ How do you solve for $m$ then? Hint: integration by substitution.

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Statistics – How to Calculate the Expected Value of a Standard Normal Distribution

You are almost there, follow your last step:

$$E[X] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} xe^{\displaystyle\frac{-x^{2}}{2}}\mathrm{d}x\\=-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-x^2/2}d(-\frac{x^2}{2})\\=-\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\mid_{-\infty}^{\infty}\\=0$$.

Or you can directly use the fact that $xe^{-x^2/2}$ is an odd function and the limits of the integral are symmetric about $x=0$.

Solved – Median of the half-Cauchy distribution

From any symmetric distribution, you can define a half-distribution simply by doubling the part of the distribution above the symmetry point (and eliminating the part below the symmetry point). Then it is clear that the upper quartile of original distribution will be the median of the half-distribution.

For the standard Cauchy this gives (using R, you can do it symbolically using the formula for the cdf):

 qcauchy(0.75)
[1] 1

Or the Cauchy with symmetry point 10 and scale parameter 2:

 qcauchy(0.75, 10, 2)
[1] 12

Best Answer

Related Solutions

Statistics – How to Calculate the Expected Value of a Standard Normal Distribution

Solved – Median of the half-Cauchy distribution

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