6 Probability Density Functions (PDFs)

CSC 411 / CSC D11 / CSC C11 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 vectorx= [x1, x2, .., xn]T. Most of the intuitions from discrete variables transferdirectly to the continuous case, although there are some describe the probabilities of a real-valued scalar variablexwith a Probability DensityFunction (PDF), writtenp(x). Any real-valued functionp(x)that satisfies:p(x) 0for allx(1) p(x)dx= 1(2)is a valid PDF. I will use the convention of upper-casePfor discrete probabilities, and lower-casepfor the PDF we can specify the Probability that the random variablexfalls within a givenrange:P(x0 x x1) = x1x0p(x)dx(3)This can be visualized by plotting the curvep(x). Then, to determine the Probability thatxfallswithin a range, we compute the area under the curve for that PDF can be thought of as the infinite limit of a discrete distribution, , a discrete dis-tribution with an infinite number of possible outcomes.

CSC 411 / CSC D11 / CSC C11 Probability Density Functions (PDFs) The off-diagonal terms are covariances: Σ ij = cov(x i,x j) = E p(x)[(x i −µ i)(x j −µ j)] (10) between variables x i and x j. If the covariance is a large positive number, then we expect x i to be largerthanµ iwhenx j islargerthanµ j ...

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