I've been asked to find the cumulative distribution function of :
$$f(x) = \begin{cases} c(1-x^2) & -1.5 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}$$
I'm attempting to understand the intuition of applying the cumulative distribution function from above question question.
I made use of following questions/answers/references :
to find the distribution function
Find C and the distribution function
https://en.wikipedia.org/wiki/Probability_density_function
The cumulative distribution function will be ? :
$$F(x)=\int_0^x{c(1-x^2)}dx$$
From the cumulative distribution function how to find find $P(-.5 \lt X \le .75)$ ?
To avoid confusion, use $F(x)=\int_0^x c(2-s)^3\mathsf d s$
Now, $\int c (2-s)^3\mathsf ds=\dfrac{-c (2-s)^4}{4}$, so
$$F(x)=\dfrac{-c}{4}{\Big[(2-s)^4\Big]}_{s=0}^{s=x}\mathbf 1_{x\in(0;\underset{?}{\tiny\boxed{3}})}+\mathbf 1_{x\in [\underset{?}{\tiny\boxed{3}};\infty)}$$
Though, as DrHab notes, the given function is negative over the latter part of its domain, so cannot be a probability density function. That 3 is likely a typographical error; perhaps it is mean to be 2?.