Someone ask me to explain the different between random variables and random process. Here are my explanation.
Say we are a wholesaler of apples in a city. One day a worker moves down a bucket of apples from a truck. By looking at the apples in this bucket, we can measure the expected weight and variation of apples in this bucket. Let denote the expected weight as , in which it is a random variable that possesses finite mean and variation.
Then, the worker moves down another bucket of apples. We not sure if this bucket of apples possess the same statistical properties as the previous bucket. By intuition, we know these apples in the second bucket possesses a finite expected weight as well (or you will see a HUGE apple!!). Next, the worker continue to move down a lot of buckets of apples from the truck (big truck!). For convenience, we denote the expected weight as where denotes the sequence where the worker moves down the buckets.
The manager of the wholesaler is not interested at the weight distribution of apples in any particular bucket but among certain groups of buckets. For example, he wants to know if the first bucket has worm, what is the chance that the nth bucket has worm as well. This question the inter-relation among the buckets. So, we have no choice but to extend the notion “a single bucket” into “groups of buckets”. In mathematics, we write this as and call this the random process (or stochastic process) of X. In such a way, we can describe the correlation (or other statistical properties) between the first bucket and the second bucket or any bucket in the later sequence to the manager.
As for the summary, in random process (stochastic process) we are dealing with a group of “homogeneous” random variables in the sense that they are of the same function (e.g. apples) but they may possess different distribution function.
Variance is one of the parameter to describe the distribution. It tells how the distribution extends from the mean value.
Moments are always used to describe the distribution due to its computational simplicity (i.e. the power series). 1st moment is mean value, 2nd moment is variance, 3rd moment is skewness.
Let and denote two random variables. Covariance is a measure of how much and change together.
In this case, variance is a special case of the covariance when two variables are identical.
Given a stochastic process , the autocovariance is the covariance of againsts itself in a time-shifting. That is .
Correlation and Autocorrelation:
Correlation is the measure of the strength of association between two variables. For example, there is a strong correlation between black eyes and and black hair as you may seem them moving together all the time. But, there is a weak correlation between blue wallpaper and puppies. Bear in mind that correlation is not causation. Though the death of babies increase with the cases of vehicle accidents. They have a strong correlation. But we cannot say the death of babies causes the happen of vehicle accidents. It does not make sense. [Taken from Yahoo! Answer]
Autocorrelation is the cross-correlation of the variable itself, in which separated by certain amount of time delay.
Say we run an experiment in which it gives a finite number of possible outcome, named as sample space .
For the convenience of mathematical calculation, each outcome (denoted as ) of is translated into a real number (if possible) and that translation process is given by a function where . We name as random variable and most of the time we just denote it as if the context is well understood.
Stationary Random Process:
Let denote a random process, in which and denote its mean value and autocorrelation function.
is non-stationary if and vary as varies.
Otherwise, is said stationary or weakly stationary in wide-sense.
For weakly stationary process, and .
For strongly stationary process, all possible moments and join moments are time-invariant.
A few thoughts after reading the article – An Early History of the Internet (Kleinrock, 2010) from IEEE Communication magazine.
Three famous pioneers in packet-switching network (i.e. a generic form of Internet) – Leonard Kleinrock (UCLA), Paul Baran (RAND) and Donald Davis (NPL, UK). Kleinrock was the student of the Shannon (Yes. He is the founder of information theory!) and his work was mainly on mathematical underpinning and simulation experiments. Meanwhile, Baran and Davis focused on the engineering and the architecture issues. However, only Kleinrock (and his colleagues) managed to roll out ARPANET (former state of Internet); while Baran failed to convince AT&T and US Air Force about his work and Davis was frustrated by the UK government.
The lessons I learned are (though you may not agree with me):
1. To success you need a good direction and a supporting agency .
2. Don’t be afraid if your work is not favored by public. (Packet switch network is weird in early days!)
3. Do maths and simulation. Math and simulation proof/results are important!
There is only a limited research actives in campus. While our research group (a.k.a G’s research group) is still small, we are not in the position to say anything. But, Dr. G always encourages us to apply for more research funding to expand the group.
He said, it is just like an chicken and egg. You have chicken and you will have eggs. Once you have eggs, you have more chickens. Meaning, once you have funding, you’ll get RA. Subsequently, you will have more research outputs and you can get another round of research grant. At the same time, G’s research group will expand and we will have more research activities.
There are two reasons that I reluctant to apply.
First, I am working on network traffic and it relies heavily on the network simulator (I presume). Yes. I do need to collect network traffic data from the networking department but I don’t think I need a specify router just for data collection. Perhaps I should request the fund be spent on the software investment e.g. OPNET and Qualnet? But, honestly, I prefer to run my own network simulator, which is much more “tractable”.
Secondly, it’ll sacrifice my “bachelor life” in research (i.e. the responsibility ….).
Masura Emota found that water crystal has life as its structure appears to change according to words or thoughts of the environmental creatures. That is to say, water crystal knows to enjoy the paint of Picasso and the music of Beethoven. He published his work entitled “Message from Water” and earns certain degree of fame among people.
But his work is not favored by research communities. (Emoto’s work is found to be a pseudo-Science)
Science is not just giving an inspiring story. Everything we claim, we need to prove.
After all, it is just an inspiring story. Not a Science.
Special thanks to THWong for introduce me the story.