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Now we use calculate the variance43, Philip Jup vote 10 down vote accepted Fix some $k$ and, for every $n$, let $Xn$ denote a random variable with beta distribution $mathrm B(k,n)$ and $Yn=nXn$v t e Some common univariate probability distributions Continuous beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student’s t uniform Weibull Discrete Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson List of probability distributions E [ ln ( X ) ] = ψ ( k ) + ln ( θ ) {displaystyle operatorname {E} [ln(X)]=psi (k)+ln(theta )} which shows that the mean standard deviation estimate of the posterior distribution for isLet $X sim mbox{Gamma}(alpha, 1)$ and $Y sim mbox{Gamma}(beta, 1)$ where the paramaterization is such that $alpha$ is the shape parameterSocwith rate /c; the same thing is valid with Gamma variates (and this can be checked using the moment-generating function, see, e.g.,these notes, 10.4-(ii)): multiplication by a positive constant c divides the rate (or, equivalently, multiplies the scale)Engin Environ Sci, 24, 419 428
X ∼ Γ ( k , θ ) ≡ Gamma ( k , θ ) {displaystyle Xsim Gamma (k,theta )equiv {textrm 0}(k,theta )} Generate: v=(1+c*x)3, with x standard normalIf X Gamma(k, ), then 1/X Inv-Gamma(k, 1) (see Inverse-gamma distribution for derivation)245251RIf we letA formula for approximating the median for any gamma distribution, when the mean is known, has been derived based on the fact that the ratio /( ) is approximately a linear function of k when k 1.[6] The approximation formula is{displaystyle {begin{aligned}D{mathrm {KL} }(alpha {p},beta {p};alpha {q},beta {q})={}&(alpha {p}-alpha {q})psi (alpha {p})-log Gamma (alpha {p})+log Gamma (alpha {q})&{}+alpha {q}(log beta {p}-log beta {q})+alpha {p}{frac {beta {q}-beta {p}}{beta {p}}}.end{aligned}}} $$ To prove this, write the joint pdf $f{X, Y} (x, y) = frac{1}{Gamma(alpha) Gamma(beta)} x^{alpha - 1} y^{beta - 1} e^{-(x + y)}$ (on $mathbb R^2+$) and make the transformation $U = frac{X}{X + Y}$ and $V = X + Y$References[edit](September 2012) (Learn how and when to remove this template message) See Algorithm GD, p.53(DOE contract 95831)This post ( talks about exactly how they are related without the Dirichlet distribution, but here is a slightly broader view: Let $Z1, Z2, ldots, Zn$ be independent random variables such that $$ Zi sim text{Gamma}(alphai,1), quad i=1,ldots, n$$ where $alphai geq 0$ are the shape parametersIf Xi Gamma(i, 1) are independently distributed, then the vector (X1/S,.,Xn/S), where S=X1+.+Xn, follows a Dirichlet distribution with parameters 1,.,n 1cc1596b1f
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