Quick question here that I cannot find in my textbook or online.
I have a probability density function as follows:
$\begin{cases} 0.04x & 0 \le x < 5 \\ 0.4 - 0.04x & 5 \le x < 10 \\ 0 & \text{otherwise} \end{cases}$
Now I understand that for the median, the value of the integral must be $0.5$. We can set the integrals from negative infinity to m where m represents the median and solve. However, there are 2 functions here so how would i do that? In the answers that I was provided, the prof. simply takes the first function and applies what I said.
How/why did he do that?
Help would be greatly appreciated! Thank you :)
I don't know, let's find out. Maybe the median is in the $[0,5]$ part. Maybe it is in the other part. To get some insight, let's find the probability that our random variable lands between $0$ and $5$. This is $$\int_0^5(0.04)x\,dx.$$ Integrate. We get $0.5$. What a lucky break! There is nothing more to do. The median is $5$.
Well, it wasn't entirely luck. Graph the density function. (We should have done that to begin with, geometric insight can never hurt.) We find that the density function is symmetric about the line $x=5$. So the median must be $5$.
Remark: Suppose the integral had turned out to be $0.4$. Then to reach the median, we need $0.1$ more in area. Then the median $m$ would be the number such that $\int_5^m (0.4-0.04x)\,dx=0.1$.