The weekly profit in thousands of dollars of Miller's Office Supply Store is random variable X whose cdf is given as follows: $F(x)=0$ for $x<0$; $F(x)=(3/32)(2x^2-x^3/3)$ for $0 \leq x \leq 4$; $F(x)=1$ for $4<x$.
(i) Find the probability of a weekly profit of at most 2000 dollars. That is, find $P(x \leq 2$).
My attempt:
I thought you would solve it using integrals like this:
$P(x \leq 2)=(3/32)$ $\int_0^2 x\ dx= (3/32)$ $[2*(x^3/9)-(x^4/12)]$
but I can not seem to get the answer. The answer is $.5$. Can someone please help me with this?
(ii) Find the probability of a weekly profit of at least 3000 dollars.
My thoughts:
For this problem, I think we can do it the same way as in (i)? If so, can someone show me? Can we also do it $P(A)=$ $1-P(x \leq 2)$. If so, how to.
(iii)Determine expressions for the pdf, $f(x)$.
I this it will be be $f(x)=0$ for $x<0$ and $x>0$. I am not sure about the other intervals.
I just started doing problems like these because I am trying to go ahead of what I already know. Can some please explain to me how would you solve the parts to this problem?
I think you need to work on obtaining a more solid, introductory understanding of the definitions of these objects you're working with.
For example, the CDF of a random variable $X$ evaluated at the number $x$ is precisely defined as the probability $$Pr(X \leq x)$$ With that in mind (i) should become a lot easier.
Note also that notation really is important, at least when first learning introductory probability theory. For example, when one speaks of the random variable $X$ one must avoid confusing $X$ with its realization, $x$.
Your subtraction idea for (ii) is exactly on the right track, but you should make sure you can justify why such a formula ($Pr(X\geq 2) = 1 - Pr(X < 2)$) actually makes sense. Hint: it has to do with adding the probability of the union of mutually exclusive events.
Again, for (iii) review the definition of the PDF and its connection to the CDF (they are related by summation/integration for discrete/continuous random variables respectively.