Find percentile of Discrete Probability Distribution

Question

This is just a hypothetical question that I had (I made this question myself, just am curious about something). What if we were trying to find the $20\text{th}$ percentile in a discrete probability distribution from $0 \leq x \leq 5$, where

$F(0) = 0.1$

$F(1) = 0.3$

$F(2) = 0.5$

$F(3) = 0.6$

$F(4) = 0.8$

$F(5) = 1$

$F(x)$ is the cdf of course.

Would the $20\text{th}$ percentile be $0$ or $1$ or something else entirely?

This is where the $20\text{th}$ percentile is the value above or equal to $20%$ of the data

Can anyone give me the algorithm to find percentile in a discrete probability distribution if there is one?

Thank you for any help/comments.

Answer 1

From comments: I think you are looking for the value $k$ where $P(X<k)≤0.2≤P(X≤k)$

• for $k=0$ you have $P(X<0)=0$ and $P(X≤0)=0.1$
• for $k=1$ you have $P(X<1)=0.1$ and $P(X≤1)=0.3$
• for $k=2$ you have $P(X<2)=0.3$ and $P(X≤2)=0.5$
• $\cdots$

so the $20\%$ percentile is at the value $1$ for this distribution.

This approach works in general, but though there may be more than one solution. If in your example you had been looking for the median $(50\%)$ then $k$ could be $2$ or $3$ or any number in between; all these solutions would then be correct, so it is an arbitrary choice - some people conventionally take the middle of the range of correct values so here $2.5$ for the median, but that too is arbitrary.