Probability of density function is greater than one

Question

I'm working on a probability function : $$\frac{x}{2}-2$$ and am trying to find the probability of a value falling between 4.5 and 7.5. I set up the equation as $$\int^{7.5}_{4.5}\frac{x}{2}-2\ dx$$ and get $$\bigg [\frac{x^2}{4} - 2x\bigg]^{7.5}_{4.5} = 3$$ which is confusing because I thought a probability couldn't be greater than 1. Would 3 be valid or am I not understanding a concept?

Answer 1

Let $f:[a,b]\to[0,\infty)$ be a probability (density) function. You know that the probability of having a value falling in the interval $[c,d]\subset [a,b]$ is given by $$\int_c^d f(x)\ dx. $$

Also, $[a,b]$ is the set of all possible values. Therefore, the integral of the probability function over all the domain is the probability of having any possible value, which has to be $1$. With this, we conclude that $$\int_a^b f(x)\ dx = 1.$$

In your case, $f(x) = \frac{x}{2}-2$. So if $[a,b]$ is the domain of $f$, it's necessary that $$\int_a^b \frac{x}{2}-2\ dx = 1.$$

If you are getting a value bigger than $1$, it's certain that you are integrating beyond the domain.

PS: if you already know $a$ then it's possible to find the bound $b$. Just substitute the value of $a$ on the integral above and solve the equation for $b$.