Let $X$ be a random variable with cumulative distribution function:
$$F(x)=\begin{cases} 0 & : x < -2 \\ 1/8 & : -2 \leq x < -1 \\ 3/8 & : -1 \leq x < 0 \\ 5/8 & : 0 \leq x < 1 \\ 7/8 & : 1 \leq x < 2 \\ 1 & : 2 \leq x \end{cases}$$
a) Find the probability distribution of X.
b) Find the expected value and std deviation of X.
c) Find the $70$th percentile.
d) Find $P(-1\leq X \leq 1)$
My work: I can't attempt b, c, and d without knowing the answer to a.
For a I'm not sure how to go about finding the probability distribution from the CDF. I know the CDF is the sum from $y\leq x$ of $f(y)$. I can't understand how to track backwards. I have looked online and on youtube for how to solve this and in all cases I've seen the cdf has functions for the value and from what I understand the cdf is the derivative of the pdf. But I just have numbers, $0, 1/8, 7/8$, the derivative of all of those is just $0$ so I don't think I can apply that to my problem.
Updated: Using @probablyme 's answer for a:
b) E(X) = 0
SD $=(4*1/8)+(1*2/8)+(0*2/8)+(1*2/8)+(4*1/8)+(0*0)=1.5$
c) $70$th percentile = $(6*0.7)=4.2$ round to $5$ thus $70$th percentile = $1/8$
d) $P(-1\leq X \leq 1)$ = F($1$) - F($-2$) = $5/8 - 0 = 5/8$
$\endgroup$ 81 Answer
$\begingroup$From the cdf, we can get the following pmf
$$f_X(x) = \begin{cases} \dfrac{1}{8}, &x = -2\\[1ex] \dfrac{2}{8}, &x = -1\\[1ex] \dfrac{2}{8}, &x = 0\\[1ex] \dfrac{2}{8}, &x = 1\\[1ex] \dfrac{1}{8}, &x = 2\\[1ex] 0, &\text{otherwise} \end{cases}$$ because the cdf only changes at $x = -2,-1,0,1,2$. Thus this is the range of $X$, and we see that $X$ is discrete.
In particular $F(-2) = f(-2) = 1/8$, $3/8 = F(-1) = f(-2)+f(-1)\implies f(-1) = 2/8$. The rest are calculated similarly.
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