Continuous random variable X has density given by f(x)=c⋅(x+1)⋅(x−3) if x∈[−1,3] and 0 otherwise.
1) Find DF(2;X) .
2) Find E[X] .

Question

Continuous random variable   X   has density given by   f(x)=c⋅(x+1)⋅(x−3)   if   x∈[−1,3]   and   0   otherwise.
1) Find   DF(2;X) . 
2) Find   E[X] .

anonymous · Answer

$$f(x)=\begin{cases}c(x+1)(x-3)&\text{for }x\in[-1,3]\[1ex]0&\text{otherwise}\end{cases}$$In order for $f(x)$ to be a valid pdf you must have
$$\int_{-\infty}^\infty f(x)\,\mathrm{d}x=c\int_{-1}^3(x+1)(x-3)\,\mathrm{d}x=1$$Finding $c$ isn't too hard.

I'm not familiar with the notation $\mathrm{DF}(2;X)$. I assume $\mathrm{DF}$ stands for "distribution function", ie. cdf, but I'm not sure what the $2;X$ is supposed to mean. Maybe the value of the cdf at $X=2$?

The expected value is simple enough once you determine the value of $c$:
$$\mathbb{E}(X)=\int_{-\infty}^\infty xf(x)\,\mathrm{d}x=c\int_{-1}^3x(x+1)(x-3)\,\mathrm{d}x=\cdots$$