Random Variables, Probability Distribution and Probability Density Function

This post is merely a note for my own reference. If you read, you will surely die of boredom :) Oh... but you can read also, in case I mess up in the definition. Okes... selamaat...

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Since my undergrad degree was in not-so-technical IT, and the masters course I took at Monash was rather light in computer science terms, I'm facing some problems with my PhD. I have dove into the world of video processing, where most of the top guns use maths., probabilities and statistical techniques as if they were eating spaghetti with meatballs. Well, at least that's what I think :)

Anywayz... today I would just like to blog about the meaning I have given to the terms Probability Distribution and Probability Density Function. I don't know why, but I seem to be mixing the two quite frequently. Not to say I'm mixing the definition, but rather mixing up the confusion regarding what I'm confused about :D Confused? Don't be, cause I already am. Anywayz....

We must first start with a Random Variable. A random variable is a variable (duh!) whose value is the outcome of a statistical experiment :) The variable itself is normally represented in UPPERCASE letters (e.g. X, or Y, or B?), and its value in LOWERCASE letters (e.g. X = x, or X = z...)

Probability Distribution - A distribution linking each outcome of a statistical experiment with its probability of occurrence (http://stattrek.com/Lesson2/ProbabilityDistribution.aspx?Tutorial=stat). Normally in the form of a table (for discrete prob. distributions) or an equation (for continuous variables). Okes... this means that... let's say the variable we are observing (X) is the TYPE OF CARD from a deck of the 4-aces. X therefore can take the values 'Ace of Hearts', 'Ace of Clubs', 'Ace of Diamonds' and 'Ace of Spades'. The statistical experiment would be to draw one card from the sub-deck consisting of the four aforementioned aces :)

Weather	Probability
AoD	0.25
AoC	0.25
AoS	0.25
AoH	0.25

So, the prob. distribution would be... as seen in the table above (where we are considering a uniform distribution and also an unbiased sub-deck). Now all these are discrete values, where X can either be one of the four at each trial. If we're talking about continuous data... then that's different, where you can't describe it using a simple table as shown above. You need an EQUATION for that... whoaaaaa!!!!

ok ok. rasa2 dah faham tu...

Now what about Probability Density Function?

Okes... a Probability Density Function (or pdf, PDF or just density function) is a function that represents the probability distribution of continuous random variables, therefore... representing also a continuous probability distribution :) And, the probability of singletons (e.g. P(X=x) = 0), since continuous means that the values are defined within a range (maybe from negative infinity to infinity... and beyond!!!)

Hmmm

ok ok. Rasa2 macam dah ok...

Selamaat