Let X be the ratio of sales to profits of some company. Assume that X has the distribution function F(x) = 0 if x

Question

Let X be the ratio of sales to profits of some company. Assume that X has the distribution function F(x) = 0 if x < 2, F(x) = (x^2-4)/5 if 2<= x < 3, F(x) = 1 if x >= 3. Find and sketch the density. What is the probability that X is between 2.5 (40% profit) and 5 (20% profit)?

anonymous · Answer

$$F(x)=\begin{cases}0&\text{for }x<2\\\frac{1}{5}(x^2-4)&\text{for }2\le x<3\\1&\text{for }x\ge3\end{cases}$$
I'll let you work on the details of the sketch.

The second part asks for $P(2.5<X<5)$, which is given by $F(5)-F(2.5)$.

Judging by the given $F(x)$, you have $F(5)=1$ and $F(2.5)=\dfrac{1}{5}(2.5^2-4)=.45$, and so $P(2.5<x<5)=.55$.