How do I myself compute the median of the following pdf?

Question

I know how to calculate the median and the third quartile of a set of data but if it’s continuous I don’t know how to compute it

Answer 1

You have to integrate from $0$ to $z$ and see where $\int_0^z f(x) \, dx =0.5$. You are looking for the value of $X=z$. Here you have to do it stepwise.

$\underline{\text{Here are the steps:}}$

1) First you to check if $\int_0^1 \frac34x^2 \, dx \geq 0.5$. If this is the case then the equation is $\int_0^z \frac34x^2 \, dx=0.5$. Calculate the integral and then solve the equation for $z$.

2) If $\int_0^1 \frac34x^2 \, dx<0.5$ then you have to use the following equation to evaluate the corresponing value of $X=z$:

$\int_0^1 \frac34x^2 \, dx+\int_{e}^{z} \frac1x \, dx =0.5$

Answer 2

In general, the median of a pdf $f:[a,b]$ is calculated by $\int_{a}^{m}{f(x) dx}=0.5$.

The integral of $3/4x^{2}$ from $0$ to $1$ is $1/4$.

So we need $\int_{e}^{m}\frac 1 x\, dx = 1/4$.

So we need $log_{e}(m)-log_{e}(e)=1/4$.

So $log(m)=1.25$ so $m=e^{1.25}$. Which is.....drumroll....3.49 :)