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Unformatted text preview: CSC 411 / CSC D11 Probability Density Functions (PDFs) 6 Probability Density Functions (PDFs) In many cases, we wish to handle data that can be represented as a real-valued random variable, or a real-valued vector x = [ x 1 ,x 2 ,...,x n ] T . Most of the intuitions from discrete variables transfer directly to the continuous case, although there are some subtleties. We describe the probabilities of a real-valued scalar variable x with a Probability Density Function (PDF), written p ( x ) . Any real-valued function p ( x ) that satisfies: p ( x ) for all x (1) integraldisplay - p ( x ) dx = 1 (2) is a valid PDF. I will use the convention of upper-case P for discrete probabilities, and lower-case p for PDFs. With the PDF we can specify the probability that the random variable x falls within a given range: P ( x x x 1 ) = integraldisplay x 1 x p ( x ) dx (3) This can be visualized by plotting the curve p ( x ) . Then, to determine the probability that x falls within a range, we compute the area under the curve for that range. The PDF can be thought of as the infinite limit of a discrete distribution, i.e., a discrete dis- tribution with an infinite number of possible outcomes. Specifically, suppose we create a discrete distribution with N possible outcomes, each corresponding to a range on the real number line. Then, suppose we increase N towards infinity, so that each outcome shrinks to a single real num- ber; a PDF is defined as the limiting case of this discrete distribution. There is an important subtlety here: a probability density is not a probability per se. For one thing, there is no requirement that p ( x ) 1 . Moreover, the probability that x attains any one specific value out of the infinite set of possible values is always zero, e.g. P ( x = 5) = integraltext 5 5 p ( x ) dx = 0 for any PDF p ( x ) . People (myself included) are sometimes sloppy in referring to p ( x ) as a probability, but it is not a probability rather, it is a function that can be used in computing probabilities. Joint distributions are defined in a natural way. For two variables x and y , the joint PDF p ( x,y ) defines the probability that ( x,y ) lies in a given domain D : P (( x,y ) D ) = integraldisplay ( x,y ) D p ( x,y ) dxdy (4) For example, the probability that a 2D coordinate ( x,y ) lies in the domain (0 x 1 , y 1) is integraltext x 1 integraltext y 1 p ( x,y ) dxdy . The PDF over a vector may also be written as a joint PDF of its variables. For example, for a 2D-vector a = [ x,y ] T , the PDF p ( a ) is equivalent to the PDF p ( x,y ) . Conditional distributions are defined as well: p ( x | A ) is the PDF over x , if the statement A is true. This statement may be an expression on a continuous value, e.g. y = 5 . As a short-hand, Copyright c circlecopyrt 2009 Aaron Hertzmann and David Fleet 27 CSC 411 / CSC D11 Probability Density Functions (PDFs) we can write p ( x | y ) , which provides a PDF for...
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- Spring '10
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