PDF Function

Returns a value from a probability density (mass) distribution.

Category:	Probability
Alias:	PMF

Syntax

PDF (dist,quantile<,parm-1, ... ,parm-k> )

Required Arguments

dist

is a character constant, variable, or expression that identifies the distribution. Valid distributions are as follows:

Distribution	Argument
Bernoulli	`Bernoulli`
Beta	`BETA`
Binomial	`BINOMIAL`
Cauchy	`CAUCHY`
Chi-Square	`CHISQUARE`
Exponential	`EXPONENTIAL`
F	`F`
Gamma	`GAMMA`
Generalized Poisson	`GENPOISSON`
Geometric	`GEOMETRIC`
Hypergeometric	`HYPERGEOMETRIC`
Laplace	`LAPLACE`
Logistic	`LOGISTIC`
Lognormal	`LOGNORMAL`
Negative binomial	`NEGBINOMIAL`
Normal	`NORMAL\|GAUSS`
Normal mixture	`NORMALMIX`
Pareto	`PARETO`
Poisson	`POISSON`
T	`T`
Tweedie	`TWEEDIE`
Uniform	`UNIFORM`
Wald (inverse Gaussian)	`WALD\|IGAUSS`
Weibull	`WEIBULL`

Note	Except for T, F, and NORMALMIX, you can minimally identify any distribution by its first four characters.

quantile

is a numeric constant, variable, or expression that specifies the value of the random variable.

Optional Argument

parm-1,...,parm-k

are optional numeric constants, variables, or expressions that specify the values of shape, location, or scale parameters that are appropriate for the specific distribution.

See	Details for complete information about these parameters.

Details

Bernoulli Distribution

PDF('BERNOULLI',x,p)

Arguments

x

is a numeric random variable.

p

is a numeric probability of success.

Range

0 ≤ p ≤ 1

Details

The PDF function for the Bernoulli distribution returns the probability density function of a Bernoulli distribution, with probability of success equal to p. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} B E R N^{'}, x, p) = {\begin{matrix} 0 & x < 0 \\ 1 - p & x = 0 \\ 0 & 0 < x < 1 \\ p & x = 1 \\ 0 & x > 1 \end{matrix} \end{matrix}

Note: There are no location or scale parameters for this distribution.

Beta Distribution

PDF('BETA',x,a,b<,l,r> )

Arguments

x

is a numeric random variable.

a

is a numeric shape parameter.

Range

a > 0

b

is a numeric shape parameter.

Range

b > 0

l

is the numeric left location parameter.

Default

r

is the right location parameter.

Default	1
Range	r > l

Details

The PDF function for the beta distribution returns the probability density function of a beta distribution, with shape parameters a and b. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} B E T A^{'}, x, a, b, l, r) = {\begin{matrix} 0 & x < l \\ \frac{1}{β (a, b)} \frac{{(x - l)}^{a - 1} {(r - x)}^{b - 1}}{{(r - l)}^{a + b - 1}} & l \leq x \leq r \\ 0 & x > r \end{matrix} \end{matrix}

Note: The quantity

\frac{x - l}{r - l}

is forced to be

ε \leq \frac{x - l}{r - l} \leq 1 - 2 ε

Binomial Distribution

PDF('BINOMIAL',m,p,n)

Arguments

m

is an integer random variable that counts the number of successes.

Range

m = 0, 1, ...

p

is a numeric probability of success.

Range

0 ≤ p ≤ 1

n

is an integer parameter that counts the number of independent Bernoulli trials.

Range

n = 0, 1, ...

Details

The PDF function for the binomial distribution returns the probability density function of a binomial distribution, with parameters p and n, which is evaluated at the value m. The equation follows:

\begin{matrix} P D F (^{'} B I N O M^{'}, m, p, n) = {\begin{matrix} 0 & m < 0 \\ (\begin{matrix} n \\ m \end{matrix}) p^{m} {(1 - p)}^{n - m} & 0 \leq m \leq n \\ 0 & m > n \end{matrix} \end{matrix}

Note: There are no location or scale parameters for the binomial distribution.

Cauchy Distribution

PDF('CAUCHY',x<,θ,λ> )

Arguments

x

is a numeric random variable.

θ

is a numeric location parameter.

Default

λ

is a numeric scale parameter.

Default	1
Range	λ > 0

Details

The PDF function for the Cauchy distribution returns the probability density function of a Cauchy distribution, with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} C A U C H Y^{'}, x, θ, λ) = \frac{1}{π} (\frac{λ}{λ^{2} + {(x - θ)}^{2}}) \end{matrix}

Chi-Square Distribution

PDF('CHISQUARE',x,df <,nc> )

Arguments

x

is a numeric random variable.

df

is a numeric degrees of freedom parameter.

Range

df > 0

nc

is an optional numeric non-centrality parameter.

Range

nc ≥ 0

Details

The PDF function for the chi-square distribution returns the probability density function of a chi-square distribution, with df degrees of freedom and non-centrality parameter nc. The PDF function is evaluated at the value x. This function accepts non-integer degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central chi-square distribution. The following equation describes the PDF function for the chi-square distribution:

\begin{matrix} P D F (^{'} C H I S Q^{'}, x, ν, λ) = {\begin{matrix} 0 & x < 0 \\ Σ_{j = 0}^{\infty} e^{- \frac{λ}{2}} \frac{{(\frac{λ}{2})}^{j}}{j!} p_{c} (x, ν + 2 j) & x \geq 0 \end{matrix} \end{matrix}

In the equation, p_c(.,.) denotes the density from the central chi-square distribution:

\begin{matrix} p_{c} (x, a) = \frac{1}{2} p_{g} (\frac{x}{2}, \frac{a}{2}) \end{matrix}

In the equation, p_g(y,b) is the density from the gamma distribution, which is given by the following equation:

\begin{matrix} p_{g} (y, b) = \frac{1}{Γ (b)} e^{- y} y^{b - 1} \end{matrix}

Exponential Distribution

PDF('EXPONENTIAL',x <,λ> )

Arguments

x

is a numeric random variable.

λ

is a scale parameter.

Default	1
Range	λ > 0

Details

The PDF function for the exponential distribution returns the probability density function of an exponential distribution, with the scale parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} E X P O^{'}, x, λ) = {\begin{matrix} 0 & x < 0 \\ \frac{1}{λ} e x p (- \frac{x}{λ}) & x \geq 0 \end{matrix} \end{matrix}

F Distribution

PDF('F',x,ndf,ddf<,nc> )

Arguments

x

is a numeric random variable.

ndf

is a numeric numerator degrees of freedom parameter.

Range

ndf> 0

ddf

is a numeric denominator degrees of freedom parameter.

Range

ddf > 0

nc

is a numeric non-centrality parameter.

Range

nc ≥ 0

Details

The PDF function for the F distribution returns the probability density function of an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and the non-centrality parameter nc. The PDF function is evaluated at the value x. This PDF function accepts non-integer degrees of freedom for ndf and ddf. If nc is omitted or equal to zero, the value returned is from a central F distribution. In the following equation, let

ν_{1} = n d f

, let

ν_{2} = d d f

, and let

λ = n c

. The equation describes the PDF function for the F distribution:

\begin{matrix} P D F (^{'} F^{'}, x, ν_{1}, ν_{2}, λ) = {\begin{matrix} 0 & x < 0 \\ Σ_{j = 0}^{\infty} e^{- \frac{λ}{2}} \frac{{(\frac{λ}{2})}^{j}}{j!} p_{f} (f, ν_{1} + 2 j, ν_{2}) & x \geq 0 \end{matrix} \end{matrix}

In the equation, p_f(f,u₁,u₂) is the density from the central F distribution:

\begin{matrix} p_{f} (f, u_{1}, u_{2}) = p_{B} (\frac{u_{1} f}{u f + u_{2}}, \frac{u_{1}}{2}, \frac{u_{2}}{2}) \frac{u_{1} u_{2}}{{(u_{2} + u_{1} f)}^{2}} \end{matrix}

In the equation p_B(x,a,b) is the density from the standard beta distribution.

Note: There are no location or scale parameters for the F distribution.

Gamma Distribution

PDF('GAMMA',x,a<,λ> )

Arguments

x

is a numeric random variable.

a

is a numeric shape parameter.

Range

a > 0

λ

is a numeric scale parameter.

Default	1
Range	λ > 0

Details

The PDF function for the gamma distribution returns the probability density function of a gamma distribution, with the shape parameter a and the scale parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} G A M M A^{'}, x, a, λ) = {\begin{matrix} 0 & x < 0 \\ \frac{1}{λ^{a} Γ (a)} x^{a - 1} e x p (- \frac{x}{λ}) & x \geq 0 \end{matrix} \end{matrix}

Generalized Poisson Distribution

PDF('GENPOISSON',x, θ, η)

Arguments

x

is an integer random variable.

θ

specifies a shape parameter.

Range

<10⁵ and >0

η

specifies a shape parameter.

Range	≥0 and <0.95
Tip	When η =0, the distribution is the Poisson distribution with a mean and variance of θ. When η>0, the mean is $θ \div (1 - η)$ and the variance is $θ \div {(1 - η)}^{3}$ .

Details

The probability mass function for the generalized Poisson distribution follows:

f (x; θ, η) = θ {(θ + η x)}^{x - 1} e^{- θ - η x} / x!, x = 0, 1, 2, ..., θ > 0, 0 \leq η < 1

Geometric Distribution

PDF('GEOMETRIC',m,p)

Arguments

m

is a numeric random variable that denotes the number of failures before the first success.

Range

m ≥ 0

p

is a numeric probability of success.

Range

0 ≤ p ≤ 1

Details

The PDF function for the geometric distribution returns the probability density function of a geometric distribution, with parameter p. The PDF function is evaluated at the value m. The equation follows:

\begin{matrix} P D F (^{'} G E O M^{'}, m, p) = {\begin{matrix} 0 & m < 0 \\ p {(1 - p)}^{m} & m \geq 0 \end{matrix} \end{matrix}

Note: There are no location or scale parameters for this distribution.

Hypergeometric Distribution

PDF('HYPER',x,N,R,n<,o> )

Arguments

x

is an integer random variable.

N

is an integer population size parameter.

Range

N = 1, 2, ...

R

is an integer number of items in the category of interest.

Range

R = 0, 1, ..., N

n

is an integer sample size parameter.

Range

n = 1, 2, ..., N

o

is an optional numeric odds ratio parameter.

Range

o > 0

Details

The PDF function for the hypergeometric distribution returns the probability density function of an extended hypergeometric distribution, with population size N, number of items R, sample size n, and odds ratio o. The PDF function is evaluated at the value x. If o is omitted or equal to 1, the value returned is from the usual hypergeometric distribution. The equation follows:

\begin{matrix} P D F & (^{'} H Y P E R^{'}, x, N, R, n, o) = \\ {\begin{matrix} 0 & x < m a x (0, R + n - N) \\ \frac{(\begin{matrix} R \\ x \end{matrix}) (\begin{matrix} N - R \\ n - x \end{matrix}) o^{x}}{Σ_{j = m a x (0, R + n - N)}^{m i n (R, n)} (\begin{matrix} R \\ j \end{matrix}) (\begin{matrix} N - R \\ n - j \end{matrix}) o^{j}} & m a x (0, R + n - N) \leq x \leq m i n (R, n) \\ 0 & x > m i n (R, n) \end{matrix} \end{matrix}

Laplace Distribution

PDF('LAPLACE',x<,θ,λ> )

Arguments

x

is a numeric random variable.

θ

is a numeric location parameter.

Default

λ

is a numeric scale parameter.

Default	1
Range	λ > 0

Details

The PDF function for the Laplace distribution returns the probability density function of the Laplace distribution, with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} L A P L A C E^{'}, x, θ, λ) = \frac{1}{2 λ} e x p (- \frac{| x - θ |}{λ}) \end{matrix}

Logistic Distribution

PDF('LOGISTIC',x<,θ,λ> )

Arguments

x

is a numeric random variable.

θ

is a numeric location parameter.

Default

λ

is a numeric scale parameter.

Default	1
Range	λ > 0

Details

The PDF function for the logistic distribution returns the probability density function of a logistic distribution, with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} L O G I S T I C^{'}, x, θ, λ) = \frac{e x p (- \frac{x - θ}{λ})}{λ {(1 + e x p (- \frac{x - θ}{λ}))}^{2}} \end{matrix}

Lognormal Distribution

PDF('LOGNORMAL',x<,θ,λ> )

Arguments

x

is a numeric random variable.

θ

specifies a numeric log scale parameter. (exp(θ) is a scale parameter.)

Default

λ

specifies a numeric shape parameter.

Default	1
Range	λ > 0

Details

The PDF function for the lognormal distribution returns the probability density function of a lognormal distribution, with the log scale parameter θ and the shape parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} L O G N^{'}, x, θ, λ) = {\begin{matrix} 0 & x \leq 0 \\ \frac{1}{x λ \sqrt[]{2 π}} e x p (- \frac{{(l o g (x) - θ)}^{2}}{2 λ^{2}}) & x > 0 \end{matrix} \end{matrix}

Negative Binomial Distribution

PDF('NEGBINOMIAL',m,p,n)

Arguments

m

is a positive integer random variable that counts the number of failures.

Range

m= 0, 1, ...

p

is a numeric probability of success.

Range

0 ≤ p ≤ 1

n

is a numeric value that counts the number of successes.

Range

n>0

Details

The PDF function for the negative binomial distribution returns the probability density function of a negative binomial distribution, with probability of success p and number of successes n. The PDF function is evaluated at the value m. The equation follows:

\begin{matrix} P D F (^{'} N E G B^{'}, m, p, n) = {\begin{matrix} 0 & m < 0 \\ (\begin{matrix} n + m - 1 \\ n - 1 \end{matrix}) p^{n} {(1 - p)}^{m} & m \geq 0 \end{matrix} \end{matrix}

Note: There are no location or scale parameters for the negative binomial distribution.

Normal Distribution

PDF('NORMAL',x<,θ,λ> )

Arguments

x

is a numeric random variable.

θ

is a numeric location parameter.

Default

λ

is a numeric scale parameter.

Default	1
Range	λ > 0

Details

The PDF function for the normal distribution returns the probability density function of a normal distribution, with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} N O R M A L^{'}, x, θ, λ) = \frac{1}{λ \sqrt[]{2 π}} e x p (- \frac{{(x - θ)}^{2}}{2 λ^{2}}) \end{matrix}

Normal Mixture Distribution

PDF('NORMALMIX',x,n,p,m,s)

Arguments

x

is a numeric random variable.

n

is the integer number of mixtures.

Range

n = 1, 2, ...

p

is the n proportions, $p_{1}, p_{2}, \dots, p_{n}$ , where $Σ_{i = 1}^{i = n} p_{i} = 1$ .

Range

p = 0, 1, ...

m

is the n means $m_{1}, m_{2}, \dots, m_{n}$ .

s

is the n standard deviations $s_{1}, s_{2}, \dots, s_{n}$ .

Range

s > 0

Details

The PDF function for the normal mixture distribution returns the probability that an observation from a mixture of normal distribution is less than or equal to x. The equation follows:

\begin{matrix} P D F (^{'} N O R M A L M I X^{'}, x, n, p, m, s) = Σ_{i = 1}^{i = n} p_{i} P D F (^{'} N O R M A L^{'}, x, m_{i}, s_{i}) \end{matrix}

Note: There are no location or scale parameters for the normal mixture distribution.

Pareto Distribution

PDF('PARETO',x,a<,k> )

Arguments

x

is a numeric random variable.

a

is a numeric shape parameter.

Range

a > 0

k

is a numeric scale parameter.

Default	1
Range	k > 0

Details

The PDF function for the Pareto distribution returns the probability density function of a Pareto distribution, with the shape parameter a and the scale parameter k. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} P A R E T O^{'}, x, a, k) = {\begin{matrix} 0 & x < k \\ \frac{a}{k} {(\frac{k}{x})}^{a + 1} & x \geq k \end{matrix} \end{matrix}

Poisson Distribution

PDF('POISSON',n,m)

Arguments

n

is an integer random variable.

Range

n= 0, 1, ...

m

is a numeric mean parameter.

Range

m > 0

Details

The PDF function for the Poisson distribution returns the probability density function of a Poisson distribution, with mean m. The PDF function is evaluated at the value n. The equation follows:

\begin{matrix} P D F (^{'} P O I S S O N^{'}, n, m) = {\begin{matrix} 0 & n < 0 \\ e^{- m} \frac{m^{n}}{n!} & n \geq 0 \end{matrix} \end{matrix}

Note: There are no location or scale parameters for the Poisson distribution.

T Distribution

PDF('T',t,df<,nc> )

Arguments

t

is a numeric random variable.

df

is a numeric degrees of freedom parameter.

Range

df > 0

nc

is an optional numeric non-centrality parameter.

Details

The PDF function for the T distribution returns the probability density function of a T distribution, with degrees of freedom df and the non-centrality parameter nc. The PDF function is evaluated at the value x. This PDF function accepts non-integer degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central T distribution. In the following equation, let

ν = d f

and let

δ = n c

\begin{matrix} P D F (^{'} T^{'}, t, ν, δ) = \frac{1}{2^{\frac{ν}{2} - 1} Γ (\frac{ν}{2})} \int_{0}^{\infty} x^{v - 1} e^{- \frac{1}{2} x^{2}} \frac{1}{\sqrt[]{2 π}} e^{- \frac{1}{2} {(\frac{t x}{\sqrt[]{v}} - δ)}^{2}} \frac{x}{\sqrt[]{v}} d x \end{matrix}

Note: There are no location or scale parameters for the T distribution.

Tweedie Distribution

PDF('TWEEDIE', y, p<,µ, φ>)

Arguments

y

is a random variable.

Range	y ≥0
Notes	This argument is required.
Notes	When y>1, y is numeric. When p=1, y is an integer.

p

is the power parameter.

Range	p ≥1
Note	This argument is required.

µ

is the mean.

Default	1
Range	µ >0

φ

is the dispersion parameter.

Default	1
Range	φ>0

Details

The PDF function for the Tweedie distribution returns an exponential dispersion model with variance and mean related by the equation variance

= ϕ * μ^{p}

The equation follows:

\frac{1}{y} Σ_{j = 1}^{\infty} (\frac{y^{- j α} ({p - 1)}^{j α}}{ϕ^{j (1 - α)} {(2 - p)}^{j} j! Γ (- j α)}) \exp (\frac{1}{ϕ} (y \frac{μ^{1 - p} - 1}{1 - p} - \frac{μ^{2 - p} - 1}{2 - p}))

The following relationship applies to the preceding equation:

α = \frac{2 - p}{1 - p}

Note: The accuracy of computed Tweedie probabilities is highly dependent on the location in parameter space. Ten digits of accuracy are usually available except when p is near 2 or

ϕ

is near 0, in which case the accuracy might be as low as six digits.

Uniform Distribution

PDF('UNIFORM',x<,l,r> )

Arguments

x

is a numeric random variable.

l

is the numeric left location parameter.

Default

r

is the numeric right location parameter.

Default	1
Range	r > l

Details

The PDF function for the uniform distribution returns the probability density function of a uniform distribution, with the left location parameter l and the right location parameter r. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} U N I F O R M^{'}, x, l, r) = {\begin{matrix} 0 & x < l \\ \frac{1}{r - l} & l \leq x \leq r \\ 0 & x > r \end{matrix} \end{matrix}

Wald (Inverse Gaussian) Distribution

PDF('WALD',x,λ<,µ>)

PDF('IGAUSS',x,λ<,µ>)

Arguments

x

is a numeric random variable.

λ

is a numeric shape parameter.

Range

λ > 0

µ

is the mean.

Default	1
Range	μ > 0

Details

The PDF function for the Wald distribution returns the probability density function of a Wald distribution, with shape parameter λ, which is evaluated at the value x. The equation follows:

f x (x) = {[\frac{λ}{{2 π x}^{3}}]}^{1 / 2} \exp {- \frac{λ}{{2 μ}^{2} x} {(x - μ)}^{2}}, x > 0

Weibull Distribution

PDF('WEIBULL',x,a<,λ> )

Arguments

x

is a numeric random variable.

a

is a numeric shape parameter.

Range

a > 0

λ

is a numeric scale parameter.

Default	1
Range	λ > 0

Details

The PDF function for the Weibull distribution returns the probability density function of a Weibull distribution, with the shape parameter a and the scale parameter λ. The PDF function is evaluated at the value x. The equation follows:

\begin{matrix} P D F (^{'} W E I B U L L^{'}, x, a, λ) = {\begin{matrix} 0 & x < 0 \\ e x p (- {(\frac{x}{λ})}^{a}) \frac{a}{λ} {(\frac{x}{λ})}^{a - 1} & x \geq 0 \end{matrix} \end{matrix}

Example

The following SAS statements produce these results.

SAS Statement	Result
y=pdf('BERN',0,.25);	0.75
y=pdf('BERN',1,.25);	0.25
y=pdf('BETA',0.2,3,4);	1.2288
y=pdf('BINOM',4,.5,10);	0.20508
y=pdf('CAUCHY',2);	0.063662
y=pdf('CHISQ',11.264,11);	0.081686
y=pdf('EXPO',1);	0.36788
y=pdf('F',3.32,2,3);	0.054027
y=pdf('GAMMA',1,3);	0.18394
y=pdf('GENPOISSON',9,1,.7);	0.0150130915
y=pdf('HYPER',2,200,50,10);	0.28685
y=pdf('LAPLACE',1);	0.18394
y=pdf('LOGISTIC',1);	0.19661
y=pdf('LOGNORMAL',1);	0.39894
y=pdf('NEGB',1,.5,2);	0.25
y=pdf('NORMAL',1.96);	0.058441
y=pdf('NORMALMIX',2.3,3,.33,.33,.34, .5,1.5,2.5,.79,1.6,4.3);	0.1166
y=pdf('PARETO',1,1);	1
y=pdf('POISSON',2,1);	0.18394
y=pdf('T',.9,5);	0.24194
y=pdf('TWEEDIE',.8,5);	0.7422908236
y=pdf('UNIFORM',0.25);	1
y=pdf('WALD',1,2);	0.56419
y=pdf('WEIBULL',1,2);	0.73576