4 Find the cumulative distribution function for the continuous probability distribution $f(x)=\frac{3 x\left(x^{2}+1\right)^{2}}{62000}$ defined on the domain $[3,7]$.
Answer
The cumulative distribution function for the given continuous probability distribution is $F(x) = \boxed{\frac{x^6}{124000} + \frac{3x^4}{124000} + \frac{3x^2}{124000} - \frac{999}{124000}}$, defined on the domain $[3, 7]$.
Step 1 ：Find the cumulative distribution function (CDF) by integrating the given probability density function (PDF) $f(x)$ over the domain $[3, x]$ where $3 \leq x \leq 7$: $F(x) = \int_{3}^{x} f(t) dt$
Step 2 ：The cumulative distribution function for the given continuous probability distribution is $F(x) = \boxed{\frac{x^6}{124000} + \frac{3x^4}{124000} + \frac{3x^2}{124000} - \frac{999}{124000}}$, defined on the domain $[3, 7]$.
