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Relationships among probability distributions
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<h2 id="special-case-of-distribution-parametrization">Special case of distribution parametrization</h2> <ul><li>A <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a> with parameters <i>n</i> = 1 and <i>p</i> is a <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a> with parameter <i>p</i>.</li> <li>A <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial distribution</a> with parameters <i>n</i> = 1 and <i>p</i> is a <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a> with parameter <i>p</i>.</li> <li>A <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a> with shape parameter <i>α</i> = 1 and rate parameter <i>β</i> is an <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> with rate parameter <i>β</i>.</li> <li>A <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a> with shape parameter <i>α</i> = <i>v</i>/2 and rate parameter <i>β</i> = 1/2 is a <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a> with <i>ν</i> <a href="/facts/Degrees_of_freedom_(statistics)/fvwKCU86"> degrees of freedom</a>.</li> <li>A <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a> with 2 degrees of freedom (<i>k</i> = 2) is an <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> with a mean value of 2 (rate <i>λ</i> = 1/2 .)</li> <li>A <a href="/facts/Weibull_distribution/1fEWxbKl">Weibull distribution</a> with shape parameter <i>k</i> = 1 and rate parameter <i>β</i> is an <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> with rate parameter <i>β</i>.</li> <li>A <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a> with shape parameters <i>α</i> = <i>β</i> = 1 is a <a href="/facts/Continuous_uniform_distribution/XbnlVljT">continuous uniform distribution</a> over the real numbers 0 to 1.</li> <li>A <a href="/facts/Beta-binomial_distribution/N8n5PpHv">beta-binomial distribution</a> with parameter <i>n</i> and shape parameters <i>α</i> = <i>β</i> = 1 is a <a href="/facts/Discrete_uniform_distribution/M52SQP5n">discrete uniform distribution</a> over the integers 0 to <i>n</i>.</li> <li>A <a href="/facts/Student%2527s_t-distribution/DeT1SDqH">Student's t-distribution</a> with one degree of freedom (<i>v</i> = 1) is a <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a> with location parameter <i>x</i> = 0 and scale parameter <i>γ</i> = 1.</li> <li>A <a href="/facts/Burr_distribution/75pvKu6P">Burr distribution</a> with parameters <i>c</i> = 1 and <i>k</i> (and scale <i>λ</i>) is a <a href="/facts/Lomax_distribution/ch9VY1mn">Lomax distribution</a> with shape <i>k</i> (and scale <i>λ</i>.)</li></ul> <h2 id="transform-of-a-variable">Transform of a variable</h2> <h3>Multiple of a random variable</h3> <p>Multiplying the variable by any positive real constant yields a scaling of the original distribution. Some are self-replicating, meaning that the scaling yields the same family of distributions, albeit with a different parameter: <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a>, <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a>, <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a>, <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a>, <a href="/facts/Erlang_distribution/dqFqlxJg">Erlang distribution</a>, <a href="/facts/Weibull_distribution/1fEWxbKl">Weibull distribution</a>, <a href="/facts/Logistic_distribution/QlQNt0uC">logistic distribution</a>, <a href="/facts/Error_distribution/UapjjPyQ">error distribution</a>, <a href="/facts/Power_law/DhAjYwwz">power-law distribution</a>, <a href="/facts/Rayleigh_distribution/vLFh4zHr">Rayleigh distribution</a>. </p><p>Example: </p> <ul><li>If <i>X</i> is a gamma random variable with shape and rate parameters (<i>α</i>, <i>β</i>), then <i>Y</i> = <i>aX</i> is a gamma random variable with parameters (<i>α</i>,<i>β</i>/<i>a</i>).</li></ul> <ul><li>If <i>X</i> is a gamma random variable with shape and scale parameters (<i>α</i>, <i>θ</i>), then <i>Y</i> = <i>aX</i> is a gamma random variable with parameters (<i>α</i>,<i>aθ</i>).</li></ul> <h3>Linear function of a random variable</h3> <p>The affine transform <i>ax</i> + <i>b</i> yields a relocation and scaling of the original distribution. The following are self-replicating: <a href="/facts/Normal_distribution/UapjjPyQ">Normal distribution</a>, <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a>, <a href="/facts/Logistic_distribution/QlQNt0uC">Logistic distribution</a>, <a href="/facts/Error_distribution/UapjjPyQ">Error distribution</a>, <a href="/facts/Power_law/DhAjYwwz">Power distribution</a>, <a href="/facts/Rayleigh_distribution/vLFh4zHr">Rayleigh distribution</a>. </p><p>Example: </p> <ul><li>If <i>Z</i> is a normal random variable with parameters (<i>μ</i> = <i>m</i>, <i>σ</i>2 = <i>s</i>2), then <i>X</i> = <i>aZ</i> + <i>b</i> is a normal random variable with parameters (<i>μ</i> = <i>am</i> + <i>b</i>, <i>σ</i>2 = <i>a</i>2<i>s</i>2).</li></ul> <h3>Reciprocal of a random variable</h3> <p>The reciprocal 1/<i>X</i> of a random variable <i>X</i>, is a member of the same family of distribution as <i>X</i>, in the following cases: <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a>, <a href="/facts/F_distribution/XNuuJtWY">F distribution</a>, <a href="/facts/Log_logistic_distribution/ze7Pasb3">log logistic distribution</a>. </p><p>Examples: </p> <ul><li>If X is a Cauchy (<i>μ</i>, <i>σ</i>) random variable, then 1/<i>X</i> is a Cauchy (<i>μ</i>/<i>C</i>, <i>σ</i>/<i>C</i>) random variable where <i>C</i> = <i>μ</i>2 + <i>σ</i>2.</li> <li>If <i>X</i> is an <i>F</i>(<i>ν</i>1, <i>ν</i>2) random variable then 1/<i>X</i> is an <i>F</i>(<i>ν</i>2, <i>ν</i>1) random variable.</li></ul> <h3>Other cases</h3> <p>Some distributions are invariant under a specific transformation. </p><p>Example: </p> <ul><li>If <i>X</i> is a beta (<i>α</i>, <i>β</i>) random variable then (1 − <i>X</i>) is a beta (<i>β</i>, <i>α</i>) random variable.</li> <li>If <i>X</i> is a binomial (<i>n</i>, <i>p</i>) random variable then (<i>n</i> − <i>X</i>) is a binomial (<i>n</i>, 1 − <i>p</i>) random variable.</li> <li>If X follows a continuous uniform distribution on [0,1], and FX(X) is its cumulative distribution function (CDF), then the random variable U=FX(X) follows a standard uniform distribution on [0,1].</li></ul> <p>Some distributions are variant under a specific transformation. </p> <ul><li>If <i>X</i> is a normal (<i>μ</i>, <i>σ</i>2) random variable then <i>e</i><i>X</i> is a lognormal (<i>μ</i>, <i>σ</i>2) random variable.</li></ul> Conversely, if <i>X</i> is a lognormal (<i>μ</i>, <i>σ</i>2) random variable then log <i>X</i> is a normal (<i>μ</i>, <i>σ</i>2) random variable. <ul><li>If <i>X</i> is an exponential random variable with mean <i>β</i>, then <i>X</i>1/<i>γ</i> is a Weibull (<i>γ</i>, <i>β</i>) random variable.</li> <li>The square of a standard normal random variable has a chi-squared distribution with one degree of freedom.</li> <li>If <i>X</i> is a <a href="/facts/Student%25E2%2580%2599s_t-distribution/DeT1SDqH">Student’s t</a> random variable with <i>ν</i> degree of freedom, then <i>X</i>2 is an <i>F</i> (1,<i>ν</i>) random variable.</li> <li>If <i>X</i> is a double exponential random variable with mean 0 and scale <i>λ</i>, then |<i>X</i>| is an exponential random variable with mean <i>λ</i>.</li> <li>A geometric random variable is the <a href="/facts/Floor_and_ceiling_functions/ssehyMMf">floor</a> of an exponential random variable.</li> <li>A <a href="/facts/Rectangular_distribution/XbnlVljT">rectangular</a> random variable is the floor of a uniform random variable.</li> <li>A <a href="/facts/Reciprocal_distribution/pEqalmXv">reciprocal</a> random variable is the exponential of a uniform random variable.</li></ul> <h2 id="functions-of-several-variables">Functions of several variables</h2> <h3>Sum of variables</h3> <p class="note">See also: <a href="/facts/List_of_convolutions_of_probability_distributions/d2MBrphK">List of convolutions of probability distributions</a></p> <p>The distribution of the sum of <a href="/facts/Independent_random_variables/NUzQtnUL">independent random variables</a> is the <a href="/facts/Convolution_of_probability_distributions/l467HgVK">convolution</a> of their distributions. Suppose Z {\displaystyle Z} is the sum of n {\displaystyle n} independent random variables X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} each with <a href="/facts/Probability_mass_function/LhurokRt">probability mass functions</a> f X i ( x ) {\displaystyle f_{X_{i}}(x)} . Then Z = ∑ i = 1 n X i . {\displaystyle Z=\sum _{i=1}^{n}{X_{i}}.} If it has a distribution from the same family of distributions as the original variables, that family of distributions is said to be <i>closed under convolution</i>. Often (always?) these distributions are also <a href="/facts/Stable_distribution/rvtRmhlz">stable distributions</a> (see also <a href="/facts/Discrete-stable_distribution/8zsgjrvd">Discrete-stable distribution</a>). </p><p>Examples of such <a href="/facts/Univariate_distribution/YjaPqYfq">univariate distributions</a> are: <a href="/facts/Normal_distribution/UapjjPyQ">normal distributions</a>, <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distributions</a>, <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distributions</a> (with common success probability), <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial distributions</a> (with common success probability), <a href="/facts/Gamma_distribution/lczcdJmw">gamma distributions</a> (with common <a href="/facts/Rate_parameter/6lxcyKUf">rate parameter</a>), <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distributions</a>, <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distributions</a>, <a href="/facts/Hyperexponential_distribution/vxa9JHSY">hyperexponential distributions</a>. </p><p>Examples:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <ul><li><ul><li>If <i>X</i>1 and <i>X</i>2 are Poisson random variables with means <i>μ</i>1 and <i>μ</i>2 respectively, then <i>X</i>1 + <i>X</i>2 is a Poisson random variable with mean <i>μ</i>1 + <i>μ</i>2.</li> <li>The sum of gamma (<i>α</i><i>i</i>, <i>β</i>) random variables has a gamma (Σ<i>α</i><i>i</i>, <i>β</i>) distribution.</li> <li>If <i>X</i>1 is a Cauchy (<i>μ</i>1, <i>σ</i>1) random variable and <i>X</i>2 is a Cauchy (<i>μ</i>2, <i>σ</i>2), then <i>X</i>1 + <i>X</i>2 is a Cauchy (<i>μ</i>1 + <i>μ</i>2, <i>σ</i>1 + <i>σ</i>2) random variable.</li> <li>If X1 and X2 are chi-squared random variables with ν1 and ν2 degrees of freedom respectively, then X1 + X2 is a chi-squared random variable with ν1 + ν2 degrees of freedom.</li> <li>If <i>X</i>1 is a normal (<i>μ</i>1, <i>σ</i>21) random variable and <i>X</i>2 is a normal (<i>μ</i>2, <i>σ</i>22) random variable, then X1 + <i>X</i>2 is a normal (<i>μ</i>1 + <i>μ</i>2, <i>σ</i>21 + <i>σ</i>22) random variable.</li> <li>The sum of <i>N</i> chi-squared (1) random variables has a chi-squared distribution with <i>N</i> degrees of freedom.</li></ul></li></ul> <p>Other distributions are not closed under convolution, but their sum has a known distribution: </p> <ul><li>The sum of <i>n</i> Bernoulli (p) random variables is a binomial (<i>n</i>, <i>p</i>) random variable.</li> <li>The sum of <i>n</i> geometric random variables with probability of success <i>p</i> is a negative binomial random variable with parameters <i>n</i> and <i>p</i>.</li> <li>The sum of <i>n</i> exponential (<i>β</i>) random variables is a gamma (<i>n</i>, <i>β</i>) random variable. Since n is an integer, the gamma distribution is also a <a href="/facts/Erlang_distribution/dqFqlxJg">Erlang distribution</a>.</li> <li>The sum of the squares of <i>N</i> standard normal random variables has a chi-squared distribution with N degrees of freedom.</li></ul> <h3>Product of variables</h3> <p>The product of independent random variables <i>X</i> and <i>Y</i> may belong to the same family of distribution as <i>X</i> and <i>Y</i>: <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a> and <a href="/facts/Log-normal_distribution/V2RXmmlE">log-normal distribution</a>. </p><p>Example: </p> <ul><li>If <i>X</i>1 and <i>X</i>2 are independent log-normal random variables with parameters (<i>μ</i>1, <i>σ</i>21) and (<i>μ</i>2, <i>σ</i>22) respectively, then <i>X</i>1 <i>X</i>2 is a log-normal random variable with parameters (<i>μ</i>1 + <i>μ</i>2, <i>σ</i>21 + <i>σ</i>22).</li></ul> <p>(See also <a href="/facts/Product_distribution/RVIMz0xc">Product distribution</a>.) </p> <h3>Minimum and maximum of independent random variables</h3> <p>For some distributions, the minimum value of several independent random variables is a member of the same family, with different parameters: <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a>, <a href="/facts/Geometric_distribution/c8wjfaoV">Geometric distribution</a>, <a href="/facts/Exponential_distribution/yY3EdV0u">Exponential distribution</a>, <a href="/facts/Extreme_value_distribution/hDG41s9y">Extreme value distribution</a>, <a href="/facts/Pareto_distribution/PgMDbLrR">Pareto distribution</a>, <a href="/facts/Rayleigh_distribution/vLFh4zHr">Rayleigh distribution</a>, <a href="/facts/Weibull_distribution/1fEWxbKl">Weibull distribution</a>. </p><p>Examples: </p> <ul><li>If <i>X</i>1 and <i>X</i>2 are independent geometric random variables with probability of success <i>p</i>1 and <i>p</i>2 respectively, then min(<i>X</i>1, <i>X</i>2) is a geometric random variable with probability of success <i>p</i> = <i>p</i>1 + <i>p</i>2 − <i>p</i>1 <i>p</i>2. The relationship is simpler if expressed in terms probability of failure: <i>q</i> = <i>q</i>1 <i>q</i>2.</li> <li>If <i>X</i>1 and <i>X</i>2 are independent exponential random variables with rate <i>μ</i>1 and <i>μ</i>2 respectively, then min(<i>X</i>1, <i>X</i>2) is an exponential random variable with rate <i>μ</i> = <i>μ</i>1 + <i>μ</i>2.</li></ul> <p>Similarly, distributions for which the maximum value of several independent random variables is a member of the same family of distribution include: <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a>, <a href="/facts/Power_law/DhAjYwwz">Power law</a> distribution. </p> <h3>Other</h3> <ul><li>If <i>X</i> and <i>Y</i> are independent standard normal random variables, <i>X</i>/<i>Y</i> is a Cauchy (0,1) random variable.</li> <li>If <i>X</i>1 and <i>X</i>2 are independent chi-squared random variables with <i>ν</i>1 and <i>ν</i>2 degrees of freedom respectively, then (<i>X</i>1/<i>ν</i>1)/(<i>X</i>2/<i>ν</i>2) is an <i>F</i>(<i>ν</i>1, <i>ν</i>2) random variable.</li> <li>If <i>X</i> is a standard normal random variable and U is an independent chi-squared random variable with <i>ν</i> degrees of freedom, then X ( U / ν ) {\displaystyle {\frac {X}{\sqrt {(U/\nu )}}}} is a Student's <i>t</i>(<i>ν</i>) random variable.</li> <li>If <i>X</i>1 is a gamma (<i>α</i>1, 1) random variable and <i>X</i>2 is an independent gamma (α2, 1) random variable then <i>X</i>1/(<i>X</i>1 + <i>X</i>2) is a beta(<i>α</i>1, <i>α</i>2) random variable. More generally, if <i>X</i>1 is a gamma(<i>α</i>1, <i>β</i>1) random variable and <i>X</i>2 is an independent gamma(<i>α</i>2, <i>β</i>2) random variable then β2 X1/(<i>β</i>2 <i>X</i>1 + <i>β</i>1 <i>X</i>2) is a beta(<i>α</i>1, <i>α</i>2) random variable.</li> <li>If <i>X</i> and <i>Y</i> are independent exponential random variables with mean μ, then <i>X</i> − <i>Y</i> is a <a href="/facts/Laplace_distribution/teMwaLry">double exponential</a> random variable with mean 0 and scale μ.</li> <li>If Xi are independent Bernoulli random variables then their <a href="/facts/Parity_function/VHXnG1uf">parity</a> (XOR) is a Bernoulli variable described by the <a href="/facts/Piling-up_lemma/XdKG4HIS">piling-up lemma</a>.</li></ul> <p>(See also <a href="/facts/Ratio_distribution/5rUEmWkG">ratio distribution</a>.) </p> <h2 id="approximate-limit-relationships">Approximate (limit) relationships</h2> <p>Approximate or limit relationship means </p> <ul><li>either that the combination of an infinite number of <i>iid</i> random variables tends to some distribution,</li> <li>or that the limit when a parameter tends to some value approaches to a different distribution.</li></ul> <p>Combination of <i>iid</i> random variables: </p> <ul><li>Given certain conditions, the sum (hence the average) of a sufficiently large number of iid random variables, each with finite mean and variance, will be approximately normally distributed. This is the <a href="/facts/Central_limit_theorem/0gc2L0Wd">central limit theorem</a> (CLT).</li></ul> <p>Special case of distribution parametrization: </p> <ul><li><i>X</i> is a hypergeometric (<i>m</i>, <i>N</i>, <i>n</i>) random variable. If <i>n</i> and <i>m</i> are large compared to <i>N</i>, and <i>p</i> = <i>m</i>/<i>N</i> is not close to 0 or 1, then <i>X</i> approximately has a Binomial(<i>n</i>, <i>p</i>) distribution.</li> <li><i>X</i> is a beta-binomial random variable with parameters (<i>n</i>, <i>α</i>, <i>β</i>). Let <i>p</i> = <i>α</i>/(<i>α</i> + <i>β</i>) and suppose <i>α</i> + <i>β</i> is large, then <i>X</i> approximately has a binomial(<i>n</i>, <i>p</i>) distribution.</li> <li>If <i>X</i> is a binomial (<i>n</i>, <i>p</i>) random variable and if <i>n</i> is large and <i>np</i> is small then <i>X</i> approximately has a Poisson(<i>np</i>) distribution.</li> <li>If <i>X</i> is a negative binomial random variable with <i>r</i> large, <i>P</i> near 1, and <i>r</i>(1 − <i>P</i>) = <i>λ</i>, then <i>X</i> approximately has a Poisson distribution with mean <i>λ</i>.</li></ul> <p>Consequences of the CLT: </p> <ul><li>If <i>X</i> is a Poisson random variable with large mean, then for integers <i>j</i> and <i>k</i>, P(<i>j</i> ≤ <i>X</i> ≤ <i>k</i>) approximately equals to <i>P</i>(<i>j</i> − 1/2 ≤ <i>Y</i> ≤ <i>k</i> + 1/2) where <i>Y</i> is a normal distribution with the same mean and variance as <i>X</i>.</li> <li>If <i>X</i> is a binomial(<i>n</i>, <i>p</i>) random variable with large <i>np</i> and <i>n</i>(1 − <i>p</i>), then for integers <i>j</i> and <i>k</i>, P(<i>j</i> ≤ <i>X</i> ≤ <i>k</i>) approximately equals to P(<i>j</i> − 1/2 ≤ <i>Y</i> ≤ <i>k</i> + 1/2) where <i>Y</i> is a normal random variable with the same mean and variance as <i>X</i>, i.e. <i>np</i> and <i>np</i>(1 − <i>p</i>).</li> <li>If <i>X</i> is a beta random variable with parameters <i>α</i> and <i>β</i> equal and large, then <i>X</i> approximately has a normal distribution with the same mean and variance, i. e. mean <i>α</i>/(<i>α</i> + <i>β</i>) and variance <i>αβ</i>/((<i>α</i> + <i>β</i>)2(<i>α</i> + <i>β</i> + 1)).</li> <li>If <i>X</i> is a gamma(<i>α</i>, <i>β</i>) random variable and the shape parameter <i>α</i> is large relative to the scale parameter <i>β</i>, then <i>X</i> approximately has a normal random variable with the same mean and variance.</li> <li>If <i>X</i> is a Student's <i>t</i> random variable with a large number of degrees of freedom <i>ν</i> then <i>X</i> approximately has a standard normal distribution.</li> <li>If <i>X</i> is an <i>F</i>(<i>ν</i>, <i>ω</i>) random variable with <i>ω</i> large, then <i>νX</i> is approximately distributed as a chi-squared random variable with <i>ν</i> degrees of freedom.</li></ul> <h2 id="compound-or-bayesian-relationships">Compound (or Bayesian) relationships</h2> <p>When one or more parameter(s) of a distribution are random variables, the <a href="/facts/Compound_probability_distribution/ccJIMBxT">compound</a> distribution is the marginal distribution of the variable. </p><p>Examples: </p> <ul><li>If <i>X</i> | <i>N</i> is a binomial (<i>N</i>,<i>p</i>) random variable, where parameter <i>N</i> is a random variable with negative-binomial (<i>m</i>, <i>r</i>) distribution, then <i>X</i> is distributed as a negative-binomial (<i>m</i>, <i>r</i>/(<i>p</i> + <i>qr</i>)).</li> <li>If <i>X</i> | <i>N</i> is a binomial (<i>N</i>,<i>p</i>) random variable, where parameter <i>N</i> is a random variable with Poisson(<i>μ</i>) distribution, then <i>X</i> is distributed as a Poisson (<i>μp</i>).</li> <li>If <i>X</i> | <i>μ</i> is a Poisson(<i>μ</i>) random variable and parameter <i>μ</i> is random variable with gamma(<i>m</i>, <i>θ</i>) distribution (where <i>θ</i> is the scale parameter), then <i>X</i> is distributed as a negative-binomial (<i>m</i>, <i>θ</i>/(1 + <i>θ</i>)), sometimes called <a href="/facts/Gamma-Poisson_distribution/znzj365Y">gamma-Poisson distribution</a>.</li></ul> <p>Some distributions have been specially named as compounds: <a href="/facts/Beta-binomial_distribution/N8n5PpHv">beta-binomial distribution</a>, <a href="/facts/Beta_negative_binomial_distribution/FdNmEqFQ">Beta negative binomial distribution</a>, <a href="/facts/Gamma-normal_distribution/cW9fI2D4">gamma-normal distribution</a>. </p><p>Examples: </p> <ul><li>If <i>X</i> is a Binomial(<i>n</i>,<i>p</i>) random variable, and parameter p is a random variable with beta(<i>α</i>, <i>β</i>) distribution, then <i>X</i> is distributed as a Beta-Binomial(<i>α</i>,<i>β</i>,<i>n</i>).</li> <li>If <i>X</i> is a negative-binomial(<i>r</i>,<i>p</i>) random variable, and parameter <i>p</i> is a random variable with beta(<i>α</i>,<i>β</i>) distribution, then <i>X</i> is distributed as a <a href="/facts/Beta_negative_binomial_distribution/FdNmEqFQ">Beta negative binomial distribution</a>(<i>r</i>,<i>α</i>,<i>β</i>).</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Central_limit_theorem/0gc2L0Wd">Central limit theorem</a></li> <li><a href="/facts/Compound_probability_distribution/ccJIMBxT">Compound probability distribution</a></li> <li><a href="/facts/List_of_convolutions_of_probability_distributions/d2MBrphK">List of convolutions of probability distributions</a></li></ul> <h2 id="external-links">External links</h2> <ul><li>Interactive graphic: <a href="http://www.math.wm.edu/~leemis/chart/UDR/UDR.html">Univariate Distribution Relationships</a></li> <li><a href="/facts/ProbOnto/fatj1kv8">ProbOnto</a> - Ontology and knowledge base of probability distributions: <a href="http://probonto.org">ProbOnto </a></li> <li><a href="http://distributome.org">Probability Distributome project includes calculators, simulators, experiments, and navigators for inter-distributional refashions and distribution meta-data</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Cook, John D. "Diagram of distribution relationships". <a href="http://www.johndcook.com/distribution_chart.html" target="_blank">http://www.johndcook.com/distribution_chart.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Dinov, Ivo D.; Siegrist, Kyle; Pearl, Dennis; Kalinin, Alex; Christou, Nicolas (2015). "Probability Distributome: a web computational infrastructure for exploring the properties, interrelations, and applications of probability distributions". Computational Statistics. 594 (2): 249–271. doi:10.1007/s00180-015-0594-6. PMC 4856044. PMID 27158191. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4856044" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4856044</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>