Interesting links to pure via integration methods
Show that \( \mathrm{f} \left( x \right) = nx^{n-1} \) will always give a pdf for the range of values where \( 0 \leq x \leq 1 \) and \( n \in \mathbb{N} \)
Show that \( \displaystyle \int^{1}_{0} \left( a-1 \right) \sum^{\infty}_{n=1} x^{a^n -1} \hspace{2px} \mathrm{d}x \) always gives a pdf for the range of values where \( a \in \mathbb{N} \) and \( a \geq 2 \)
I have recently been tasked with delivering professional development (PD) around statistics – primarily binomial distributions. When delivering PD, I always want to make sure that we have a ‘pathway’ from fundamental basic concepts into the topic being focussed on and then the trajectory beyond, due in part to the basic understanding that the basic concepts (if correctly conceived) allow for the natural development of all areas along this learning structure.
Therefore, I’ve looked very carefully at the concept of probability distributions as the grounding concept for statistics and, in particular, pdfs (probability density functions). The two questions above have arisen naturally from this examination as when one is setting questions, one is naturally trying to find definite integrals of functions which have a value of: 1. When searching this way for questions, one usually finds ‘obvious facts’, which once recognised, makes one go, ‘Oh of course – why didn’t I recognise that as something obvious!’.
Perhaps it’s just me, and everyone else already has done so, but just in case you haven’t, the above two questions revolve around one such instance. Obvious fact: Integrating a basic polynomial expression between 0 and 1 will always give a unit fraction. Therefore we can actually define unit fractions as the value for the area under the curve of a polynomial expression, i.e.:
\[ \displaystyle \frac{1}{n} = \int^{1}_{0} x^{n-1} \hspace{2px} \mathrm{d}x \] Note then that we are emphasising that we can consider the fraction to be the area as an object, allowing us to think of things in terms of areas and scalings. So first of all, if we scale all of these areas up to 1 by multiplying by ‘\( n \)’, i.e.: \[ \displaystyle \int^{1}_{0} nx^{n-1} \hspace{2px} \mathrm{d}x = 1 \] Then we can see we have a generalised set of p.d.f.’s for all ‘\( n \)’ (where \( n \in \mathbb{N} \)). This is potentially useful because if students are aware of this generalisation, then if an exam question is set on a specific p.d.f. it might be highly likely that one of these would be used, and students are now familiar with the generalisation. So we may hope that a single specific instance won’t throw them. It’s also a pleasant realisation to have about integrating polynomials. [As an exercise, sketch graph the above to familiarise yourself with the concepts]. The next question plays with some obvious facts around infinite series, combined with the previous observation. The implication of place value structures with a base of ‘\( a \)’ is that: \[ \displaystyle \sum^{\infty}_{n=1} \frac{a-1}{a^n} = 1 \hspace{20px} \mathrm{for} \hspace{3px} a\in \mathbb{N}, \hspace{2px} \geq 2 \] Therefore, we use these infinite series to define unit fractions by simply factorising out \((a-1)\) and rearranging: \[ \displaystyle \frac{1}{a-1} = \sum^{\infty}_{n=1} \frac{1}{a^n} \]Without spelling it out, if you carefully consider these statements and the initial question regarding thinking in terms of areas, you should see how the second question was naturally constructed.
by Jeremy Dawson