How To Find Distribution Function From Density Function

Function whose integral over a region describes the probability of an event occurring in that region

Geometric visualisation of the mode, median and mean of an arbitrary probability density function.^[one]

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a role whose value at any given sample (or point) in the sample infinite (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would exist close to that sample.^{[2] [iii]} In other words, while the absolute likelihood for a continuous random variable to have on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at 2 different samples can be used to infer, in whatever detail draw of the random variable, how much more likely information technology is that the random variable would be close to ane sample compared to the other sample.

In a more precise sense, the PDF is used to specify the probability of the random variable falling within a item range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the surface area under the density function but above the horizontal axis and betwixt the lowest and greatest values of the range. The probability density part is nonnegative everywhere, and its integral over the unabridged infinite is equal to one.

The terms "probability distribution function"^[4] and "probability function"^[5] have as well sometimes been used to announce the probability density office. However, this utilize is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over full general sets of values or information technology may refer to the cumulative distribution function, or information technology may be a probability mass function (PMF) rather than the density. "Density function" itself is likewise used for the probability mass function, leading to further defoliation.^[6] In general though, the PMF is used in the context of detached random variables (random variables that have values on a countable set), while the PDF is used in the context of continuous random variables.

Example [edit]

Suppose bacteria of a certain species typically live 4 to 6 hours. The probability that a bacterium lives exactly 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, merely there is no take a chance that whatever given bacterium dies at exactly 5.00... hours. However, the probability that the bacterium dies betwixt five hours and five.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., two%). So, the probability that the bacterium dies between 5 hours and 5.001 hours should be nigh 0.002, since this time interval is i-10th equally long as the previous. The probability that the bacterium dies betwixt 5 hours and 5.0001 hours should be most 0.0002, and so on.

In this example, the ratio (probability of dying during an interval) / (elapsing of the interval) is approximately constant, and equal to ii per hour (or 2 hour⁻¹). For example, there is 0.02 probability of dying in the 0.01-hour interval betwixt 5 and five.01 hours, and (0.02 probability / 0.01 hours) = two hour^−ane. This quantity 2 hour⁻¹ is called the probability density for dying at effectually 5 hours. Therefore, the probability that the bacterium dies at five hours can exist written as (2 hour⁻¹) dt. This is the probability that the bacterium dies within an infinitesimal window of time effectually 5 hours, where dt is the duration of this window. For example, the probability that it lives longer than v hours, simply shorter than (5 hours + 1 nanosecond), is (2 hour⁻¹)×(1 nanosecond) ≈ 6×10^−xiii (using the unit of measurement conversion 3.6×10¹² nanoseconds = i hour).

There is a probability density function f with f(five hours) = 2 hour⁻¹. The integral of f over any window of fourth dimension (not simply infinitesimal windows but also large windows) is the probability that the bacterium dies in that window.

Absolutely continuous univariate distributions [edit]

A probability density function is about commonly associated with absolutely continuous univariate distributions. A random variable ${\displaystyle Ten}$ has density ${\displaystyle f_{X}}$ , where ${\displaystyle f_{X}}$ is a non-negative Lebesgue-integrable function, if:

${\displaystyle \Pr[a\leq X\leq b]=\int _{a}^{b}f_{X}(ten)\,dx.}$

Hence, if ${\displaystyle F_{Ten}}$ is the cumulative distribution part of ${\displaystyle 10}$ , and then:

${\displaystyle F_{X}(x)=\int _{-\infty }^{x}f_{X}(u)\,du,}$

and (if ${\displaystyle f_{X}}$ is continuous at ${\displaystyle ten}$ )

${\displaystyle f_{10}(x)={\frac {d}{dx}}F_{10}(x).}$

Intuitively, one tin think of ${\displaystyle f_{X}(ten)\,dx}$ as beingness the probability of ${\displaystyle X}$ falling within the infinitesimal interval ${\displaystyle [10,x+dx]}$ .

Formal definition [edit]

(This definition may be extended to whatsoever probability distribution using the measure-theoretic definition of probability.)

A random variable ${\displaystyle X}$ with values in a measurable space ${\displaystyle ({\mathcal {10}},{\mathcal {A}})}$ (usually ${\displaystyle \mathbb {R} ^{n}}$ with the Borel sets as measurable subsets) has as probability distribution the measure X _∗ P on ${\displaystyle ({\mathcal {X}},{\mathcal {A}})}$ : the density of ${\displaystyle 10}$ with respect to a reference measure ${\displaystyle \mu }$ on ${\displaystyle ({\mathcal {X}},{\mathcal {A}})}$ is the Radon–Nikodym derivative:

${\displaystyle f={\frac {dX_{*}P}{d\mu }}.}$

That is, f is any measurable function with the property that:

${\displaystyle \Pr[X\in A]=\int _{X^{-1}A}\,dP=\int _{A}f\,d\mu }$

for whatsoever measurable set ${\displaystyle A\in {\mathcal {A}}.}$

Give-and-take [edit]

In the continuous univariate case above, the reference measure is the Lebesgue measure out. The probability mass office of a discrete random variable is the density with respect to the counting measure over the sample space (ordinarily the gear up of integers, or some subset thereof).

It is not possible to define a density with reference to an arbitrary measure out (due east.grand. ane can't choose the counting mensurate every bit a reference for a continuous random variable). Furthermore, when it does exist, the density is almost unique, meaning that any two such densities coincide almost everywhere.

Further details [edit]

Unlike a probability, a probability density office can take on values greater than i; for example, the compatible distribution on the interval [0, 1/2] has probability density f(ten) = two for 0 ≤x ≤ 1/2 and f(x) = 0 elsewhere.

The standard normal distribution has probability density

${\displaystyle f(x)={\frac {1}{\sqrt {two\pi }}}\;e^{-x^{2}/2}.}$

If a random variable Ten is given and its distribution admits a probability density office f, so the expected value of X (if the expected value exists) can be calculated as

${\displaystyle \operatorname {East} [X]=\int _{-\infty }^{\infty }10\,f(x)\,dx.}$

Not every probability distribution has a density function: the distributions of discrete random variables exercise not; nor does the Cantor distribution, even though it has no discrete component, i.due east., does non assign positive probability to whatever individual point.

A distribution has a density function if and merely if its cumulative distribution function F(x) is absolutely continuous. In this case: F is nigh everywhere differentiable, and its derivative can be used equally probability density:

${\displaystyle {\frac {d}{dx}}F(x)=f(x).}$

If a probability distribution admits a density, and then the probability of every one-point set {a} is zero; the same holds for finite and countable sets.

Two probability densities f and g represent the same probability distribution precisely if they differ merely on a set up of Lebesgue measure cypher.

In the field of statistical physics, a non-formal reformulation of the relation higher up between the derivative of the cumulative distribution part and the probability density function is generally used as the definition of the probability density part. This alternate definition is the following:

If dt is an infinitely small-scale number, the probability that Ten is included within the interval (t,t +dt) is equal to f(t)dt, or:

${\displaystyle \Pr(t<Ten<t+dt)=f(t)\,dt.}$

Link between discrete and continuous distributions [edit]

It is possible to represent certain detached random variables equally well as random variables involving both a continuous and a discrete part with a generalized probability density part, by using the Dirac delta function. (This is not possible with a probability density part in the sense defined higher up, it may be washed with a distribution.) For case, consider a binary discrete random variable having the Rademacher distribution—that is, taking −i or 1 for values, with probability ½ each. The density of probability associated with this variable is:

${\displaystyle f(t)={\frac {1}{2}}(\delta (t+1)+\delta (t-1)).}$

More more often than not, if a discrete variable can have n different values among real numbers, then the associated probability density role is:

${\displaystyle f(t)=\sum _{i=i}^{n}p_{i}\,\delta (t-x_{i}),}$

where ${\displaystyle x_{1}\ldots ,x_{n}}$ are the discrete values accessible to the variable and ${\displaystyle p_{1},\ldots ,p_{northward}}$ are the probabilities associated with these values.

This substantially unifies the treatment of discrete and continuous probability distributions. For case, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability...

Families of densities [edit]

It is common for probability density functions (and probability mass functions) to be parametrized—that is, to be characterized by unspecified parameters. For example, the normal distribution is parametrized in terms of the hateful and the variance, denoted by ${\displaystyle \mu }$ and ${\displaystyle \sigma ^{2}}$ respectively, giving the family of densities

${\displaystyle f(x;\mu ,\sigma ^{2})={\frac {1}{\sigma {\sqrt {ii\pi }}}}eastward^{-{\frac {1}{ii}}\left({\frac {x-\mu }{\sigma }}\correct)^{2}}.}$

It is important to go on in heed the difference between the domain of a family of densities and the parameters of the family. Different values of the parameters describe different distributions of different random variables on the same sample infinite (the same set of all possible values of the variable); this sample space is the domain of the family of random variables that this family unit of distributions describes. A given set of parameters describes a single distribution within the family unit sharing the functional class of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain merely parameters, but not variables, are part of the normalization factor of a distribution (the multiplicative gene that ensures that the area nether the density—the probability of something in the domain occurring— equals 1). This normalization gene is outside the kernel of the distribution.

Since the parameters are constants, reparametrizing a density in terms of dissimilar parameters, to requite a label of a dissimilar random variable in the family, ways simply substituting the new parameter values into the formula in place of the old ones. Irresolute the domain of a probability density, however, is trickier and requires more piece of work: meet the department below on change of variables.

Densities associated with multiple variables [edit]

For continuous random variables Ten ₁, …, X_n , it is too possible to define a probability density function associated to the set equally a whole, often called joint probability density function. This density function is divers as a role of the due north variables, such that, for any domain D in the n-dimensional infinite of the values of the variables Ten ₁, …, X_n , the probability that a realisation of the set variables falls inside the domain D is

${\displaystyle \Pr \left(X_{ane},\ldots ,X_{n}\in D\correct)=\int _{D}f_{X_{i},\ldots ,X_{northward}}(x_{one},\ldots ,x_{n})\,dx_{1}\cdots dx_{n}.}$

If F(x ₁, …,x _n ) = Pr(X ₁ ≤10 _ane, …,X _n  ≤10 _northward ) is the cumulative distribution role of the vector (X ₁, …,10 _n ), then the articulation probability density function can be computed equally a partial derivative

${\displaystyle f(x)=\left.{\frac {\partial ^{n}F}{\partial x_{1}\cdots \partial x_{n}}}\correct|_{x}}$

Marginal densities [edit]

For i = ane, ii, …, n, let f _{X i} (x _i ) be the probability density function associated with variable X_i alone. This is called the marginal density part, and can be deduced from the probability density associated with the random variables 10 _i, …, X_n by integrating over all values of the other due north − one variables:

${\displaystyle f_{X_{i}}(x_{i})=\int f(x_{1},\ldots ,x_{north})\,dx_{1}\cdots dx_{i-1}\,dx_{i+1}\cdots dx_{n}.}$

Independence [edit]

Continuous random variables X ₁, …, Ten_n admitting a articulation density are all independent from each other if and only if

${\displaystyle f_{X_{i},\ldots ,X_{north}}(x_{1},\ldots ,x_{n})=f_{X_{1}}(x_{1})\cdots f_{X_{n}}(x_{n}).}$

Corollary [edit]

If the joint probability density function of a vector of due north random variables can exist factored into a product of n functions of one variable

${\displaystyle f_{X_{1},\ldots ,X_{n}}(x_{i},\ldots ,x_{n})=f_{1}(x_{1})\cdots f_{northward}(x_{northward}),}$

(where each f_i is non necessarily a density) then the northward variables in the set are all independent from each other, and the marginal probability density function of each of them is given by

${\displaystyle f_{X_{i}}(x_{i})={\frac {f_{i}(x_{i})}{\int f_{i}(ten)\,dx}}.}$

Instance [edit]

This elementary example illustrates the higher up definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call ${\displaystyle {\vec {R}}}$ a 2-dimensional random vector of coordinates (Ten, Y): the probability to obtain ${\displaystyle {\vec {R}}}$ in the quarter plane of positive x and y is

${\displaystyle \Pr \left(X>0,Y>0\correct)=\int _{0}^{\infty }\int _{0}^{\infty }f_{X,Y}(x,y)\,dx\,dy.}$

Function of random variables and modify of variables in the probability density function [edit]

If the probability density function of a random variable (or vector) X is given as f_X (ten), it is possible (but oftentimes not necessary; see below) to calculate the probability density function of some variable Y = 1000(10). This is too chosen a "change of variable" and is in practice used to generate a random variable of arbitrary shape f _g(X) = f_Y using a known (for instance, uniform) random number generator.

It is tempting to think that in society to find the expected value E(g(Ten)), one must first find the probability density f _g(X) of the new random variable Y = g(X). However, rather than calculating

${\displaystyle \operatorname {Eastward} {\big (}g(10){\big )}=\int _{-\infty }^{\infty }yf_{k(X)}(y)\,dy,}$

ane may find instead

${\displaystyle \operatorname {E} {\big (}g(X){\big )}=\int _{-\infty }^{\infty }g(ten)f_{Ten}(x)\,dx.}$

The values of the two integrals are the same in all cases in which both X and yard(X) actually have probability density functions. It is not necessary that g exist a ane-to-ane function. In some cases the latter integral is computed much more easily than the former. See Law of the unconscious statistician.

Scalar to scalar [edit]

Let ${\displaystyle g:{\mathbb {R} }\rightarrow {\mathbb {R} }}$ be a monotonic function, and so the resulting density function is

${\displaystyle f_{Y}(y)=f_{Ten}{\big (}g^{-1}(y){\big )}\left|{\frac {d}{dy}}{\big (}k^{-1}(y){\big )}\right|.}$

Here grand ⁻¹ denotes the inverse role.

This follows from the fact that the probability contained in a differential expanse must exist invariant nether change of variables. That is,

${\displaystyle \left|f_{Y}(y)\,dy\correct|=\left|f_{X}(x)\,dx\correct|,}$

or

${\displaystyle f_{Y}(y)=\left|{\frac {dx}{dy}}\right|f_{X}(x)=\left|{\frac {d}{dy}}(10)\right|f_{X}(ten)=\left|{\frac {d}{dy}}{\big (}chiliad^{-i}(y){\big )}\correct|f_{Ten}{\large (}g^{-ane}(y){\large )}={\left|\left(g^{-ane}\right)'(y)\correct|}\cdot f_{X}{\big (}k^{-ane}(y){\big )}.}$

For functions that are not monotonic, the probability density part for y is

${\displaystyle \sum _{chiliad=1}^{north(y)}\left|{\frac {d}{dy}}g_{1000}^{-1}(y)\right|\cdot f_{10}{\big (}g_{1000}^{-1}(y){\big )},}$

where n(y) is the number of solutions in ten for the equation ${\displaystyle one thousand(x)=y}$ , and ${\displaystyle g_{k}^{-1}(y)}$ are these solutions.

Vector to vector [edit]

Suppose x is an n-dimensional random variable with joint density f. If y = H(10), where H is a bijective, differentiable role, then y has density g:

${\displaystyle k(\mathbf {y} )=f{\Bigl (}H^{-1}(\mathbf {y} ){\Bigr )}\left|\det \left[\left.{\frac {dH^{-i}(\mathbf {z} )}{d\mathbf {z} }}\correct|_{\mathbf {z} =\mathbf {y} }\right]\right|}$

with the differential regarded as the Jacobian of the changed of H(⋅), evaluated at y .^[7]

For example, in the 2-dimensional case x = (x ₁,x ₂), suppose the transform H is given as y ₁ = H ₁(x ₁,x _ii), y ₂ = H _two(x ₁,x _ii) with inverses x ₁ = H _ane ^−one(y ₁,y _two), 10 _two = H ₂ ⁻¹(y _ane,y ₂). The joint distribution for y = (y ₁, y₂) has density^[8]

${\displaystyle g(y_{1},y_{2})=f_{X_{1},X_{2}}{\big (}H_{1}^{-1}(y_{ane},y_{2}),H_{2}^{-1}(y_{ane},y_{ii}){\big )}\left\vert {\frac {\partial H_{1}^{-1}}{\partial y_{1}}}{\frac {\partial H_{2}^{-ane}}{\partial y_{2}}}-{\frac {\partial H_{1}^{-1}}{\partial y_{2}}}{\frac {\partial H_{2}^{-one}}{\partial y_{1}}}\right\vert .}$

Vector to scalar [edit]

Let ${\displaystyle V:{\mathbb {R} }^{n}\rightarrow {\mathbb {R} }}$ be a differentiable function and ${\displaystyle X}$ be a random vector taking values in ${\displaystyle {\mathbb {R} }^{north}}$ , ${\displaystyle f_{X}}$ be the probability density function of ${\displaystyle X}$ and ${\displaystyle \delta (\cdot )}$ be the Dirac delta role. It is possible to use the formulas in a higher place to determine ${\displaystyle f_{Y}}$ , the probability density role of ${\displaystyle Y=Five(X)}$ , which will be given by

${\displaystyle f_{Y}(y)=\int _{\mathbb {R} ^{n}}f_{Ten}(\mathbf {x} )\delta {\big (}y-Five(\mathbf {x} ){\large )}\,d\mathbf {ten} .}$

This result leads to the police force of the unconscious statistician:

${\displaystyle \operatorname {E} _{Y}[Y]=\int _{\mathbb {R} }yf_{Y}(y)\,dy=\int _{\mathbb {R} }y\int _{\mathbb {R} ^{n}}f_{X}(\mathbf {x} )\delta {\big (}y-V(\mathbf {ten} ){\big )}\,d\mathbf {x} \,dy=\int _{{\mathbb {R} }^{due north}}\int _{\mathbb {R} }yf_{Ten}(\mathbf {ten} )\delta {\big (}y-V(\mathbf {ten} ){\big )}\,dy\,d\mathbf {x} =\int _{\mathbb {R} ^{n}}V(\mathbf {x} )f_{Ten}(\mathbf {x} )\,d\mathbf {x} =\operatorname {E} _{X}[V(10)].}$

Proof:

Allow ${\displaystyle Z}$ be a collapsed random variable with probability density function ${\displaystyle p_{Z}(z)=\delta (z)}$ (i.due east. a constant equal to zero). Let the random vector ${\displaystyle {\tilde {Ten}}}$ and the transform ${\displaystyle H}$ be defined as

${\displaystyle H(Z,X)={\brainstorm{bmatrix}Z+V(X)\\Ten\end{bmatrix}}={\begin{bmatrix}Y\\{\tilde {X}}\stop{bmatrix}}.}$

Information technology is articulate that ${\displaystyle H}$ is a bijective mapping, and the Jacobian of ${\displaystyle H^{-1}}$ is given by:

${\displaystyle {\frac {dH^{-1}(y,{\tilde {\mathbf {x} }})}{dy\,d{\tilde {\mathbf {x} }}}}={\brainstorm{bmatrix}1&-{\frac {dV({\tilde {\mathbf {x} }})}{d{\tilde {\mathbf {x} }}}}\\\mathbf {0} _{northward\times i}&\mathbf {I} _{n\times n}\finish{bmatrix}},}$

which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is one. Applying the change of variable theorem from the previous section nosotros obtain that

${\displaystyle f_{Y,X}(y,x)=f_{X}(\mathbf {x} )\delta {\large (}y-Five(\mathbf {10} ){\large )},}$

which if marginalized over ${\displaystyle x}$ leads to the desired probability density part.

Sums of independent random variables [edit]

The probability density function of the sum of ii independent random variables U and 5, each of which has a probability density role, is the convolution of their carve up density functions:

${\displaystyle f_{U+V}(x)=\int _{-\infty }^{\infty }f_{U}(y)f_{V}(x-y)\,dy=\left(f_{U}*f_{V}\right)(x)}$

It is possible to generalize the previous relation to a sum of N independent random variables, with densities U ₁, …, U_North :

${\displaystyle f_{U_{ane}+\cdots +U_{Northward}}(x)=\left(f_{U_{i}}*\cdots *f_{U_{Northward}}\right)(x)}$

This tin be derived from a two-way change of variables involving Y=U+5 and Z=V, similarly to the instance below for the quotient of independent random variables.

Products and quotients of independent random variables [edit]

Given 2 contained random variables U and V, each of which has a probability density function, the density of the product Y =UV and quotient Y=U/Five can be computed by a alter of variables.

Example: Quotient distribution [edit]

To compute the caliber Y =U/V of 2 independent random variables U and 5, ascertain the post-obit transformation:

${\displaystyle Y=U/V}$

${\displaystyle Z=V}$

Then, the joint density p(y,z) can exist computed by a modify of variables from U,V to Y,Z, and Y tin can be derived by marginalizing out Z from the articulation density.

The inverse transformation is

${\displaystyle U=YZ}$

${\displaystyle V=Z}$

The absolute value of the Jacobian matrix determinant ${\displaystyle J(U,Five\mid Y,Z)}$ of this transformation is:

${\displaystyle \left|\mathrm {det} {\begin{bmatrix}{\frac {\partial u}{\partial y}}&{\frac {\partial u}{\fractional z}}\\{\frac {\partial v}{\partial y}}&{\frac {\fractional v}{\partial z}}\finish{bmatrix}}\correct|=\left|\mathrm {det} {\begin{bmatrix}z&y\\0&i\finish{bmatrix}}\correct|=|z|.}$

Thus:

${\displaystyle p(y,z)=p(u,v)\,J(u,v\mid y,z)=p(u)\,p(v)\,J(u,5\mid y,z)=p_{U}(yz)\,p_{V}(z)\,|z|.}$

And the distribution of Y can be computed past marginalizing out Z:

${\displaystyle p(y)=\int _{-\infty }^{\infty }p_{U}(yz)\,p_{5}(z)\,|z|\,dz}$

This method crucially requires that the transformation from U,5 to Y,Z be bijective. The above transformation meets this because Z can exist mapped directly back to V, and for a given V the quotient U/Five is monotonic. This is similarly the instance for the sum U +V, departure U −V and product UV.

Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables.

Example: Quotient of 2 standard normals [edit]

Given two standard normal variables U and Five, the quotient tin can exist computed every bit follows. Showtime, the variables have the following density functions:

${\displaystyle p(u)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {u^{2}}{2}}}}$

${\displaystyle p(5)={\frac {1}{\sqrt {ii\pi }}}e^{-{\frac {v^{ii}}{ii}}}}$

We transform as described above:

${\displaystyle Y=U/5}$

${\displaystyle Z=Five}$

This leads to:

${\displaystyle {\brainstorm{aligned}p(y)&=\int _{-\infty }^{\infty }p_{U}(yz)\,p_{V}(z)\,|z|\,dz\\[5pt]&=\int _{-\infty }^{\infty }{\frac {one}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}y^{2}z^{2}}{\frac {ane}{\sqrt {2\pi }}}e^{-{\frac {one}{2}}z^{2}}|z|\,dz\\[5pt]&=\int _{-\infty }^{\infty }{\frac {1}{2\pi }}e^{-{\frac {1}{2}}\left(y^{2}+1\right)z^{2}}|z|\,dz\\[5pt]&=2\int _{0}^{\infty }{\frac {one}{2\pi }}e^{-{\frac {1}{2}}\left(y^{two}+1\right)z^{2}}z\,dz\\[5pt]&=\int _{0}^{\infty }{\frac {1}{\pi }}e^{-\left(y^{2}+ane\correct)u}\,du&&u={\tfrac {1}{2}}z^{2}\\[5pt]&=\left.-{\frac {1}{\pi \left(y^{2}+ane\right)}}e^{-\left(y^{2}+one\right)u}\right|_{u=0}^{\infty }\\[5pt]&={\frac {1}{\pi \left(y^{2}+1\right)}}\cease{aligned}}}$

This is the density of a standard Cauchy distribution.

Meet also [edit]

• Density estimation
• Kernel density estimation
• Likelihood function
• List of probability distributions
• Probability amplitude
• Probability mass function
• Secondary mensurate
• Uses every bit position probability density:
  • Atomic orbital
  • Home range

References [edit]

1. ^ "AP Statistics Review - Density Curves and the Normal Distributions". Archived from the original on 2 April 2015. Retrieved 16 March 2015.
2. ^ Grinstead, Charles 1000.; Snell, J. Laurie (2009). "Provisional Probability - Discrete Conditional" (PDF). Grinstead & Snell's Introduction to Probability. Orange Grove Texts. ISBN161610046X . Retrieved 2019-07-25 .
3. ^ "probability - Is a uniformly random number over the real line a valid distribution?". Cross Validated . Retrieved 2021-10-06 .
4. ^ Probability distribution office PlanetMath Archived 2011-08-07 at the Wayback Motorcar
5. ^ Probability Function at MathWorld
6. ^ Ord, J.K. (1972) Families of Frequency Distributions, Griffin. ISBN 0-85264-137-0 (for example, Table v.ane and Example 5.4)
7. ^ Devore, Jay L.; Berk, Kenneth North. (2007). Modern Mathematical Statistics with Applications. Cengage. p. 263. ISBN0-534-40473-one.
8. ^ David, Stirzaker (2007-01-01). Elementary Probability. Cambridge University Printing. ISBN0521534283. OCLC 851313783.

Further reading [edit]

• Billingsley, Patrick (1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons. ISBN0-471-00710-ii.
• Casella, George; Berger, Roger Fifty. (2002). Statistical Inference (2d ed.). Thomson Learning. pp. 34–37. ISBN0-534-24312-vi.
• Stirzaker, David (2003). Elementary Probability . ISBN0-521-42028-8. Capacity 7 to 9 are most continuous variables.

External links [edit]

• Ushakov, N.G. (2001) [1994], "Density of a probability distribution", Encyclopedia of Mathematics, Ems Press
• Weisstein, Eric West. "Probability density part". MathWorld.

Source: https://en.wikipedia.org/wiki/Probability_density_function

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