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I am trying to work out and plot the CDF from a PDF. The pdf is $$ f(x) = \begin{cases} 0&\text{if}\quad -1\leq x \leq 1\\ |1/x^3|&\text{ otherwise} \end{cases}. $$

I know that the CDF is the integral of the PDF, and I tried integrating the PDF but all I got was 1, and I'm not sure how to plot this.

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    $\begingroup$ In general $F(x)=\int_{-\infty}^xf(u)du$ where $f$ denotes PDF and $F$ denotes CDF. Maybe you only computed $\int_{-\infty}^{\infty}f(u)du$ ("...all i got was $1$..."). $\endgroup$
    – drhab
    Oct 28, 2022 at 8:44
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    $\begingroup$ Are you able to find $F(x)$ by doing the integration proposed in my comment? $\endgroup$
    – drhab
    Oct 28, 2022 at 10:17

1 Answer 1

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Note that yours is not a valid probability density function.

Indeed, you have $f(x)< 0$, for all $x< -1$. But, by definition, a PDF is always non-negative, ie. $$f(x)\geq 0, \text{ for all } x.$$

In addition,

$$ \int_{-\infty}^{-1} 1/x^3\,dx + \int_{1}^{\infty} 1/x^3\, dx= -\frac{1}{2}+\frac{1}{2}=0. $$ Thus the CDF is also not valid either.

Update

Note: The above post refers to the original question. Below is the answer to the updated question.

For this pdf the CDF is

$$ F(x) = \begin{cases} \frac{1}{2x^2}& \text{ if } x\leq -1\\ \frac{1}{2} & \text{ if } x\in[0,1]\\ 1-\frac{1}{2x^2}& \text{ if } x>1 \end{cases}. $$

Below you can see the PDF, the CDF and the relative R code.

enter image description here enter image description here

ff_pdf <- function(x) abs(1/x^3)
ff_cdf_l <- function(x) (1/(2*x^2))
ff_cdf_r <- function(x) 1/2 + 1/2 - 1/(2*x^2)

x_l <- seq(-7,-1,len=50)
x_r <- seq(1,7,len=50)
pdfv_l <- sapply(x_l, ff_pdf)
pdfv_r <- sapply(x_r, ff_pdf)


plot(x_l, pdfv_l, type = "l", xlim=c(-7,7),
     lwd=2, ylim=c(0,1.3), ylab = "Density", xlab="x")
points(x_r, pdfv_r, type = "l", xlim=c(-7,7), lwd=2)
segments(x0=-1, y0=0, x1=1, y1=0, lwd=2)


cdfv_l <- sapply(x_l, ff_cdf_l)
cdfv_r <- sapply(x_r, ff_cdf_r)


plot(x_l, cdfv_l, type = "l", xlim=c(-7,7),
     lwd=2, ylim=c(0,1.1), ylab = "Distribution function", xlab="x")
segments(x0=-1, y0=1/2, x1=1, y1=1/2, lwd=2)
points(x_r, cdfv_r, type = "l",lwd=2)
abline(h = c(0,1), lwd=2, lty=2, col="gray")
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    $\begingroup$ This is swift. You ought to check the definition before even proceeding. +1. $\endgroup$ Oct 28, 2022 at 9:31
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    $\begingroup$ @henry This answers your original question. Later your question was edited and function $f$ is now a PDF. $\endgroup$
    – drhab
    Oct 28, 2022 at 10:07
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    $\begingroup$ @henry I've updated my answer. $\endgroup$
    – utobi
    Oct 28, 2022 at 13:26
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    $\begingroup$ @henry, if you find this answer helpful, please upvote and accept it by clicking on the tick symbol alongside the post. $\endgroup$ Oct 29, 2022 at 6:36

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