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# PDF

 PDFgives the probability density function for the symbolic distribution dist evaluated at x. PDFgives the multivariate probability density function for a symbolic distribution dist evaluated at . PDF[dist]gives the PDF as a pure function.
• For continuous distributions, PDF[dist, x] dx gives the probability that an observed value will lie between x and for infinitesimal dx.
• For discrete distributions, PDF gives the probability that an observed value will be x.
• For continuous multivariate distributions, PDF[dist, {x1, x2, ...}]dx1 dx2 ... gives the probability that an observed value will lie in the box given by the limits and for infinitesimal .
• For discrete multivariate distributions, PDF gives the probability that an observed value will be .
The PDF of a univariate continuous distribution:
The PDF of a univariate discrete distribution:
The PDF of a multivariate continuous distribution:
The PDF for a multivariate discrete distribution:
The PDF of a univariate continuous distribution:
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The PDF of a univariate discrete distribution:
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The PDF of a multivariate continuous distribution:
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The PDF for a multivariate discrete distribution:
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 Scope   (18)
Obtain exact numeric results:
Obtain a machine-precision result:
Obtain a result at any precision for a continuous distribution:
Obtain a result at any precision for a discrete distribution with inexact parameters:
PDF threads element-wise over lists:
Multivariate distributions:
PDF for non-parametric distributions:
Compare with the value for the underlying parametric distribution:
Plot the PDF for a histogram distribution:
Closed form expression for the PDF of a kernel mixture distribution:
Plot of the PDF of a bivariate smooth kernel distribution:
Product of independent distributions:
Component mixture distribution:
Quadratic transformation of a discrete distribution:
Censored distribution:
Truncated distribution:
Parameter mixture distribution:
Copula distribution:
Formula distributions defined by its PDF:
Defined by its CDF:
Defined by its survival function:
Marginal distribution:
 Applications   (10)
Plot a continuous PDF:
Plot a discrete PDF:
Plot a continuous bivariate PDF:
Plot a discrete bivariate PDF:
Plot a family of univariate continuous PDFs:
Compute the CDF from the PDF by solving a differential equation:
Plot a confidence interval for a standard normal distribution:
Compute the mode of a distribution from its PDF:
Compute the PDF after an affine transformation:
Verify the Poisson approximation of the binomial distribution for large and small :
The integral or sum over the support of the distribution is unity:
The CDF is the integral of the PDF for continuous distributions; :
The CDF is the integral of the PDF ; :
The CDF is the sum of the PDF for discrete distributions :
The survival function is the integral of the PDF ; :
The expectation for for a continuous distribution is the PDF-weighted integral :
The expectation for for a discrete distribution is the PDF-weighted sum :
The probability of for a discrete univariate distribution is given by the PDF:
The hazard function of a distribution is a ratio of the PDF and the survival function:
Symbolic closed forms do not exist for some distributions:
Numerical evaluation works:
Substitution of invalid values into symbolic outputs can give results that are not meaningful:
Passing it as an argument will generate correct results:
PDF for a truncated binormal distribution:
Isosurfaces for a trivariate normal distribution:
Isosurfaces for PDF when varying a correlation coefficient:
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