Median Value + Mode for Hybrid Functions of a Continuous Probability Density Function

Question

To find the median:

should I set the integral to 0.5.... but because there are two functions that are non-zero, I am unaware of a method to find the median.

To find the mode:

would I need to derive each function that is non-zero, set this to zero and solve for x. Then sub the values of x back into f(x) for each non-zero function. I will select the x value that produced the largest f(x).

Answer 1

The area (probability) up to $0$ is $(1)(0.2)$, which is $0.2$. To get to area $0.5$, we need another $0.3$. So if $m$ is the median then $$\int_0^m (0.2+1.2x)\,dx=0.3.$$ The rest is calculation. We could also solve the problem geometrically.

For the mode, graph the density function. It reaches its maximum at $1$.

Answer 2

Hint (to find median):$$0.5=\int_m^{\infty}f(x)dx=\int_m^1(0.2+1.2x)dx$$

Answer 3

Calculating the Mode:

You can see, that the function in the three intervals has no relative extremum. Thus you have to examine the bounderies.