Probability — Question 5

Question 5 – Solution

Step 1: Identify the reflection. $$f(1-x)=2(1-x)=g(x).$$ Therefore, the graph of $g$ is the reflection of the graph of $f$ across $x=1/2$.

Step 2: Find the mean of $f$ geometrically. The area under $f(x)=2x$ is a right triangle with vertices $$(0,0),\qquad (1,0),\qquad (1,2).$$ The $x$-coordinate of a triangle’s centroid is the average of its vertex $x$-coordinates: $$\mu_f=\frac{0+1+1}{3}=\frac23.$$

Step 3: Reflect the mean. Reflection across $x=1/2$ sends a location $x$ to $1-x$. Hence $$\mu_g=1-\mu_f=1-\frac23=\frac13.$$ $$\boxed{\mu_f=\frac23,\qquad \mu_g=\frac13}$$ As a check, the two reflected means add to $1$.