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PDF
✖
PDF

✖
PDF[dist,x]

gives the probability density function for the distribution dist evaluated at x.

✖
PDF[dist,{x1,x2,…}]

gives the multivariate probability density function for a distribution dist evaluated at {x1,x2,…}.

✖
PDF[dist]

gives the PDF as a pure function.

Details

  • For discrete distributions, PDF is also known as a probability mass function.
  • For continuous distributions, PDF[dist,x] dx gives the probability that an observed value will lie between x and x+dx for infinitesimal dx.
  • For discrete distributions, PDF[dist,x] 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 xi and xi+dxi for infinitesimal dxi.
  • For discrete multivariate distributions, PDF[dist,{x1,x2,…}] gives the probability that an observed value will be {x1,x2,…}.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

The PDF of a univariate continuous distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-crvabh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-cubfzx
Out[2]=2

The PDF of a univariate discrete distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-ibu4tk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-j5du1
Out[2]=2

The PDF of a multivariate continuous distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-cifkrj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-njup1w
Out[2]=2

The PDF for a multivariate discrete distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-od3dae
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-j1nznt
Out[2]=2

Scope  (23)Survey of the scope of standard use cases

Parametric Distributions  (5)

Obtain exact numeric results:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-dgckxh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-f8gibm
Out[2]=2

Obtain a machine-precision result:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-kdtpo7
Out[1]=1

Obtain a result at any precision for a continuous distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-d6m9eu
Out[1]=1

Obtain a result at any precision for a discrete distribution with inexact parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-e696
Out[1]=1

PDF threads element-wise over lists:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-uxh42
Out[1]=1

Multivariate distributions:

In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-mcr1m6
Out[2]=2

Nonparametric Distributions  (4)

PDF for non-parametric distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-bhbipu
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-ezf28g
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-fnodal
Out[3]=3

Compare with the value for the underlying parametric distribution:

In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-llcjdj
Out[4]=4

Plot the PDF for a histogram distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-bezwuv
Out[1]=1

Closed-form expression for the PDF of a kernel mixture distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-dldvgo
Out[1]=1

Plot of the PDF of a bivariate smooth kernel distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-d1pyr5
Out[1]=1

Derived Distributions  (10)

Product of independent distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-8pyz0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-i185f
Out[2]=2

Component mixture distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-g8oy2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-l97u83
Out[2]=2

Quadratic transformation of a discrete distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-qx2dx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-k0a3dq
Out[2]=2

Censored distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-d2ukan
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-59tyl
Out[2]=2

Truncated distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-b8ebyy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-bnm6za
Out[2]=2

Parameter mixture distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-bct2he
Out[1]=1

Copula distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-o4vib6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-cng2z2
Out[2]=2

Formula distribution defined by its PDF:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-few7li
Out[1]=1

Defined by its CDF:

In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-ceb6ni
Out[2]=2

Defined by its survival function:

In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-hfkb2
Out[3]=3

Marginal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-gmxz4v
Out[1]=1

The PDF for QuantityDistribution assumes the argument is a Quantity with compatible units:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-z6cqh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-bjwd0p
Out[2]=2

This allows for direct quantity substitution:

In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-bs6882
Out[3]=3

Compare with the direct use of the quantity argument:

In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-kkvt7w
Out[4]=4

Random Processes  (4)

Find the PDF for a SliceDistribution of a discrete-state random process:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-hha5jz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-ec193c
Out[2]=2

A continuous-state random process:

In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-cg4akz
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-dzmz38
Out[4]=4

Find the multiple time-slice PDF for a discrete-state process:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-r98gn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-hlbkqt
Out[2]=2

A multi-slice for a continuous-state process:

In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-h5vy9j
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-m37pj
Out[4]=4

Find the PDF for the StationaryDistribution of a discrete-state random process:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-mpszqf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-bdqzil
Out[2]=2

Find the slice distribution for time :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-h72hjq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-e0fhml
Out[2]=2

Applications  (10)Sample problems that can be solved with this function

Visualizing PDFs  (5)

Plot a continuous PDF:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-f0ov1z
Out[1]=1

Plot a discrete PDF:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-btvbs
Out[1]=1

Plot a continuous bivariate PDF:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-f22pm2
Out[1]=1

Plot a discrete bivariate PDF:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-f59enu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-cgdrmm
Out[2]=2

Plot a family of univariate continuous PDFs:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-i7zu82
Out[1]=1

Computing the CDF  (1)

Compute the CDF from the PDF by solving a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-cwkmw8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-bnwt3y
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-em6kvj
Out[3]=3

Confidence Intervals  (1)

Plot a confidence interval for a standard normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-buka0l
Out[1]=1

Compute boundaries of the 70% confidence interval:

In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-b3ko5c
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-o4537x
Out[3]=3

Mode of a Distribution  (1)

Compute the mode of a distribution from its PDF:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-d9w7hc
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-d9h69e
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-bnpqq7
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-fjgoo
Out[4]=4

Affine Transformations  (1)

Compute the PDF after an affine transformation:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-ec05s1
Out[1]=1

Poisson Approximation to Binomial  (1)

Verify the Poisson approximation of the binomial distribution for large and small :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-b9c2b0
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-pnmfu
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-drjm46
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-gyrntf
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bn6enu-bzo9yq
Out[5]=5

Properties & Relations  (9)Properties of the function, and connections to other functions

The integral or sum over the support of the distribution is unity:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-irh6rs
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-dwmrz2
Out[2]=2

The CDF is the integral of the PDF for continuous distributions; :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-nrya2e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-ii9ir5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-ev6jjf
Out[3]=3

The CDF is the integral of the PDF ; :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-ls1vpz
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-dzq36n
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-o923w
In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-jyp266
Out[4]=4

The CDF is the sum of the PDF for discrete distributions :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-fnsjkn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-hd6k6l
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-jych9e
Out[3]=3

The survival function is the integral of the PDF ; :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-hps8vg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-bid0zg
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-kdflkq
Out[3]=3

Expectation for for a continuous distribution is the PDF-weighted integral :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-bjga3w
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-dlhu9o
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-ci8ibw
Out[3]=3

The expectation for for a discrete distribution is the PDF-weighted sum :

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-cenlaj
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-hogpc5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-lt6w3
Out[3]=3

The probability of for a discrete univariate distribution is given by the PDF:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-bztb39
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-k67x0r
Out[2]=2

The HazardFunction of a distribution is a ratio of the PDF and the survival function:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-epuujk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-gbf1mf
Out[2]=2

Possible Issues  (3)Common pitfalls and unexpected behavior

Symbolic closed forms do not exist for some distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-gtite
Out[1]=1

Numerical evaluation works:

In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-k4p5g
Out[2]=2

Substitution of invalid values into symbolic outputs can give results that are not meaningful:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-gk0wnb
Out[1]=1

Passing it as an argument will generate correct results:

In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-d88ikd
Out[2]=2

The PDF of a distribution whose measure is incompatible with the Lebesgue measure or counting measure on the integer lattice may not evaluate or may give an incorrect result:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-foy3vo

The result of the PDF is not normalized:

In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-gk9aqa
Out[2]=2

The distribution measure has an atom at the origin, and hence is incompatible with the Lebesgue measure:

In[3]:=3
✖
https://wolfram.com/xid/0bn6enu-coz5ws
Out[3]=3

The incompatibility manifests itself in a jump discontinuity of the CDF at the atom location:

In[4]:=4
✖
https://wolfram.com/xid/0bn6enu-dbwu
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bn6enu-ggfyc
Out[5]=5

Mixed distributions are fully supported by Expectation, Probability, RandomVariate, etc.:

In[6]:=6
✖
https://wolfram.com/xid/0bn6enu-cor556
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0bn6enu-fcf0lv
Out[7]=7

Neat Examples  (3)Surprising or curious use cases

PDF for a truncated binormal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-bsv6sj
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-c8a4wj
Out[2]=2

Isosurfaces for a trivariate normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-b2fz20
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-jwdkdd
Out[2]=2

Isosurfaces for PDF when varying a correlation coefficient:

In[1]:=1
✖
https://wolfram.com/xid/0bn6enu-cm6fau
In[2]:=2
✖
https://wolfram.com/xid/0bn6enu-bh19ht
Out[2]=2
Wolfram Research (2007), PDF, Wolfram Language function, https://reference.wolfram.com/language/ref/PDF.html (updated 2010).
✖
Wolfram Research (2007), PDF, Wolfram Language function, https://reference.wolfram.com/language/ref/PDF.html (updated 2010).

Text

Wolfram Research (2007), PDF, Wolfram Language function, https://reference.wolfram.com/language/ref/PDF.html (updated 2010).

✖
Wolfram Research (2007), PDF, Wolfram Language function, https://reference.wolfram.com/language/ref/PDF.html (updated 2010).

CMS

Wolfram Language. 2007. "PDF." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2010. https://reference.wolfram.com/language/ref/PDF.html.

✖
Wolfram Language. 2007. "PDF." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2010. https://reference.wolfram.com/language/ref/PDF.html.

APA

Wolfram Language. (2007). PDF. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PDF.html

✖
Wolfram Language. (2007). PDF. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PDF.html

BibTeX

@misc{reference.wolfram_2025_pdf, author="Wolfram Research", title="{PDF}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PDF.html}", note=[Accessed: 25-May-2025 ]}

✖
@misc{reference.wolfram_2025_pdf, author="Wolfram Research", title="{PDF}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PDF.html}", note=[Accessed: 25-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_pdf, organization={Wolfram Research}, title={PDF}, year={2010}, url={https://reference.wolfram.com/language/ref/PDF.html}, note=[Accessed: 25-May-2025 ]}

✖
@online{reference.wolfram_2025_pdf, organization={Wolfram Research}, title={PDF}, year={2010}, url={https://reference.wolfram.com/language/ref/PDF.html}, note=[Accessed: 25-May-2025 ]}