I am viewing a video from MIT OpenCourseWare. The course name is Discrete Stochastic Processes, Course number 6.262.
In the second video lecture, at minute 46:53, the ratio of the same Binomial Random Variable was explored. The purpose is to show the probability density function of the Binomial Random Variable increases and decreases for certain values of the input. I agree with this, from this specific derivation. However, I am curious about some of the specific mathematical statements.
From the attached image, ,
it can be seen that $ \frac{\mathrm{P}(k+1)}{\mathrm{P}(k)} = \frac{n-k}{k+1} \frac{p}{q}\ldots (\text{1})$
$p$ and $q = 1-p$ are real values in the set $[0,1]$. $n$ is fixed, and $0 \leq k \leq n$. It appears useful to set the ratio in equation (1) either greater than or less than 1, to show the increasing / decreasing nature of $\mathrm{P}(k)$. So doing this, I get:
$ \frac{\mathrm{P}(k+1)}{\mathrm{P}(k)} < 1 \text{ when } k > np-q$ $ \frac{\mathrm{P}(k+1)}{\mathrm{P}(k)} > 1 \text{ when } k < np-q$
It is at this point that I am unable to verify the statements in the slides. How do I reach the middle part, with the three cases? My cases don't exactly match up with the cases on the slides.
When the ratio $\frac{f(k+1)}{f(k)}$ is greater than 1 and function $f$ is non-negative, it means the value of the function at 1 unit ahead is greater than the function at the current position. It could be the case that the function is increasing, yet the ratio is showing a value less than 1 (if we're nearing the top of the hill).
How was the chart plotted? I see the notation for the floor of the value $pn$, but I was unable to reach that.
Lastly, how was the final statement reached? I think the floor function is part of the difficulty, and the general quantities $q$, $p$, which are between 0 and 1 (which would tend to agree with use of the floor function).
Thanks.