Moment Generating Functions - Binomial

The last of the moment generating functions for the three main distributions. the Binomial Distribution is as follows:

Any comments or questions just add a comment.

Moment Generating Functions - Poisson

As I did for the Normal Distribution, I will show how you can derive the moment generating function for the Poisson Distribution:

Again, if you have any questions about the LaTeX or mathematics used, please just comment.

Time signatures

One thing I really love in music, but just don't see enough of, is interesting time signatures. I'm not saying that a good ol' 4/4 or 2/4 are dreary, but when an obscure band whips up a crazy sound in 26/4 it just adds another element to the music, it brings out something that would normally be missed. This might sound stupid but let me explain. 

A lot of bands would use drums to just keep the rhythm and rely on other elements of the song, such as AC/DC, to create the sound the artists are after. I'm sorry if you're a fan but just flick through Black Ice, the drums in every song boil down to dun, tss, dun, tss, dun, tss, with minor variations (granted, with the exception of the last track, Black Ice, which is a little more lively). Now, sure enough you can tap along to this music and it's very easy to listen to, but if you really want to push your listening experience, you have to move into something with a bit more vigour and technicality. A good example of making good use of a set of drums would be Tool's Lateralus when the vocals start at around 1:40. Rather than keeping the rhythm alone, it also helps to tell the story, making it a much more satisfying experience because everything in the song works to submerge the listener into it, in the same way that a film does. Instead of actors working in a white room, the background is brought to life to convey a mood, a setting, part of the story. Something we'd not usually notice directly is what's doing a lot of the work to get the audience in the best place to experience the whole product.

This is why time signatures have a certain something about them that can be exploited to add many more layers of sounds to a song. As we usually don't think about them, we don't notice that they seldom vary much in mainstream music, we're happy tapping along to the same basic rhythm because we're too easily pleased by the superficial aspects - primarily the melody followed very, very closely by the vocals and lyrics, then by the sexually suggestive dances and facial expressions. It seems a lot of chart-topping stuff is they same template repeated over and over, because that is what works, it's what people like. It's a shame that most of these people wouldn't want to stray into music that mixes things up because you can't dance to it, and it takes a little more work to get the full experience. But I assure you, there's a whole other world to music if you're willing to put the effort in, and the enjoyment you get out of it will be ten-fold what it is with today's most popular.

Anyway, back to time signatures. I'm not an expert on the matter but I can understand the basics, which has been enough for me to pick out some more interesting rhythms. For example, Meshuggah's (who have built up a reputation for odd time signatures) "Future Breed Machine" (http://www.youtube.com/watch?v=vOd-T58qHLA), at around 3:40 in has a solid 7/4 for the guitar laid onto a 4/4 drums. The drums follow the basic tss tss tss dun tss tss tss dun... and the guitars follow two lots of three beats, which is some kind of djent chugs, followed by one beat of rest. I count it as:

1, 2, 3 - 1, 2, 3 - 1 -  1, 2, 3 - 1, 2, 3 - 1. 

This may be a little more difficult to grasp at first, but don't worry it gets obscenely worse. If the previous song seemed easy enough to follow, have a listen to "Rational Gaze" (http://www.youtube.com/watch?v=rkrjE4QRsys) which has been said to follow a 23/16 during the second part of the intro, from around 0:29. If you're trying to count this and follow the beats, try counting it as three groups of 7/16 and one of 2/16, so:

1,2,3,4,5,6,7,  1,2,3,4,5,6,7,  1,2,3,4,5,6,7,  1,2.

I have yet to find out myself or from anywhere what the rest is timed in. Regardless, I think I can fully appreciate it. One way to get the most out of one of these songs is to listen different instruments and try to zones out others. For example, if the drums are very prominent in a song, try to listen more to the bass guitarist. Maybe he's playing a different rhythm or is bouncing between the lead guitar and the drummer, enhancing both at the same time. Once you've done this to all the instruments, go back and listen to it as a whole and you'll see that there is much more to some (albeit, rarely chart-toppers) songs that you would've never noticed. 

This may seem to be approaching the limit of complexity that a mere human can identify, but there's one more level - math rock. These guys know how to knock out something that will leave you in awe of it's sophistication and technicality. Some of them border on the random...some are random, but there are some that have written their music and sheets and made tabs and all the rest, so it these guys that are the masters. A brilliant song with a very interesting signature is "26 is dancer than 4" by This Town Needs Guns (http://www.youtube.com/watch?v=IhPljk5EZ0Q). Here the song title acknowledges the life that more complicated rhythms breath into music. It says that 26/4 has more feel and can please the listener more than a 4/4. Yes, that does mean:

1,2,3,4,...,24,25,26.

each measure (but there is only one from the start to the first chorus). It's a slow beat in the song dispite the pace of the melody, taking from the start until around 23 seconds in to count the 26 beats. 

Okay, I lied, there is another level (maybe two). I've a few pieces of music that offer an other worldly approach to music and seek only to demonstrate how the pitches and volume of notes and tones, and their distances apart, at the most fundamental level can give a very rewarding experience to the listener, if only s/he has the patience to find it. Some examples are Stockhausen's Klavierstück I - X (especially IX) which feature a very interesting approach to music, as well as incorporating very odd time signatures, including a mind melting 142/8. Returning to math rock, there is Tera Melos. Give them a listen and see what you think. They've passed the point where humans can tap their foot to feel the rhythm, I think if you tired tapping your foot along to "A Spoonful of Slurry" you're likely to give yourself and injury. However, to those brave souls that do, I will offer a KitKat (fo' serious) as a reward to anyone that can figure out all of the time signatures for this song, first one to comment with the correct signatures wins. God speed and good luck.

I thought someone should point out...

The irony of chinos: blokes wear them to look cool and to impress the ladies, but just end up looking gay or like Olly Murs.

The irony of tube mini skirts: women wear them to look good and feel confident, but just end up looking wider at the middle and needy.

...not that it's my place to say. You can do what they want, but you can't expect me to like it. Granted most if not all will not expect me to, but as they don't care what I think, this post shouldn't bother them. If it does and you feel like commenting, congratulations for voicing your opinion...just like me. Good day.

...on religion

If there's ever a time for a more philosophical blog then it is now, while I'm sitting in the late hours with Beethoven's 9th playing, sipping on Laphroaig out of a gorgeous pewter hip-flask (courtesy of a very, very special person).

As a man of science, or more specifically mathematics, people are usually taken aback when I express anything other than pure distaste towards any religious ideologies. Now don't get the wrong idea, I'm not what most people would describe as a theist but I have thought a lot about what exactly it is all about. I tend to avoid the subject because it's a potential minefield of political correctness. I find the best way, if you really want to sit down and think about religion on the whole, is to argue both sides fairly, if not biased towards what you're arguing against. I will try to do just that.  To start with I will make it clear that I'm comparing two groups of people; those who believe in a 'higher power', and those who don't. One of the main problems with this is that theists view themselves as superior to atheists. They feel like they have the ability to have faith or that they have broadened their minds to see beyond, on the same token, atheists look down at their rivals, seeing them as gullible, closed-minded, and insecure. This is seen their arguments for their own position such as, an atheist would argue that if a higher power or 'God' exists, then we should be able to find him/her/it, we should be able to find evidence. This instantly becomes an argument of swings and roundabouts. Both think they're better, but for reason that don't apply to the opposing group. Such an atheist would ask theists for evidence of God, and as these theists wouldn't be reliant on the scientific method, they would say something along the lines one of two things:

1. Evidence is not relevant. That an all-powerful being can easily evade all human attempts to track Him down, should He not want to be found. This seems like a childish attempt to avoid answering the difficult questions, but you have to put yourself in their shoes, believe that the being they describe exists. They believe this to be the absolute truth, which undercuts all that science could throw at it. How do you measure something that can appear to not even exist? From here all sorts of questions about the motives of God can be brought in. Why is He hiding? Does He show Himself to anyone? Why won't He reveal Himself to remove all doubt? But remember, such an all-powerful God would be infinitely more intelligent than any human, so it's pointless to question Him. 'Evidence is not relevant' is a check-mate response that shouldn't be used because it leads nowhere towards a better understanding for either parties.

2. You're the scientists, why don't you prove God doesn't exist? Seems fair, if the scientific method works then God will be disproven or proven outright, or if it doesn't, all remains a mystery, right? Except it's not that simple. I remember a nice analogy for the problems when proving the existence or non-existence of God: imagine a teapot orbiting the sun at some arbitrary distance. No one would've ever seen it or know of it's existence, but regardless of this, a man pipes up one day and claims just that. His says there is a teapot orbiting the sun. Soon enough the populating splits in two, those who believe and those who don't. The theists of the teapot ask the non-believers to prove that there is no teapot if they're so confident. This task would mean searching every cubic foot of the space around the sun and not finding anything, but an atheist responds by saying that if the teapot exists it would be much easier for one the believers to point it out. Clearly here, the teapot represents a God, but the arguments are strikingly similar. These would be, crudely speaking, the atheists asking the theists to use an atheist method for proving the theists are correct. Whereas the theists would be asking the atheists to use a method they don't believe in to prove an idea they don't believe in. Something isn't right. Neither group can prove to the other group their beliefs because they don't use a universally agreed method.

I believe that so far science is on track. It has made countless discoveries that would appear to be independent of a greater intelligence, whilst remaining true to form in its method. But the simple fact is that, even if science finally believes it knows everything, it may still be missing out on one thing - the higher power that doesn't want to be found.

This is just a brief introduction to my opinions, but discussions, counter-arguments, general comments are welcome and encouraged.

Accidental maths

I'm not sure of the uses of the following equation, but either way it's quite novel and the proof quite nice because in mathematics the factorial function, f(x) = x!, can be quite frustrating sometimes. Using it often results in using the Gamma function which is getting into mathematics quite a bit beyond Joe Bloggs. Anyway, after stumbling across the following and proving it, then realising someone had already done it and getting over the disappointment of finding out I hadn't discovered something new in maths, I've written a nice, neat version of my findings:

Again, if you have any questions on the maths used or if you like to know more about the LaTeX used, or have anything suggestions on extensions or improvements of this proof, just leave a comment.

The One.

You know you've found the one when she'd travel around the world just to be with you, but would never have to, because you'd already be on your way to be with her.