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# expectation of log gamma distribution

For integer values of $n$, Did Star Trek ever tackle slavery as a theme in one of its episodes? $$J(\lambda, n) = \frac{\lambda^n}{\Gamma(n)} \int_0^\infty t^{n-1} e^{-\lambda t} \log(1+t)\ dt = \frac{1}{\Gamma(n)} \int_0^\infty s^{n-1} e^{-s} \log(1+s/\lambda) \ ds$$, $$\Psi \left( n \right) -\ln \left( \lambda \right) +{\frac { Conditional probability of the sum of r.v. which seems to be correct when n is a non-integer. whuber (2018): "What is the expected value of the logarithm of Gamma distribution?" Roll an N sided die K times. Could someone please break it down for me. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. How to sustain this sedentary hunter-gatherer society? To learn more, see our tips on writing great answers. Quick link too easy to remove after installation, is this a problem? MathOverflow is a question and answer site for professional mathematicians. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.$$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. It is defined It only takes a minute to sign up. What's the current state of LaTeX3 (2020)? Thanks. probability expectation gamma-function gamma-distribution Theorem: Let $X$ be a random variable following a gamma distribution: Then, the expectation of the natural logarithm of $X$ is, Proof: Let $Y = \ln(X)$, such that $\mathrm{E}(Y) = \mathrm{E}(\ln X)$ and consider the special case that $b = 1$. Thanks for contributing an answer to Mathematics Stack Exchange! The delta method gives $$\operatorname{E}[f(X)] \approx f(\operatorname{E}[X]) + \frac{f''(\operatorname{E}[X])}{2} \operatorname{Var}[X]$$ so for $f(x) = \log \Gamma(x)$ it is not difficult to obtain the approximation $$\operatorname{E}[\log \Gamma(X)] \approx \log \Gamma(k \theta) + \frac{1}{2} k\theta^2 \psi^{(1)} (k\theta),$$ where $\psi^{(1)}$ is the second derivative of the logarithm of the gamma function. Note that the integrand in \eqref{eq:Y-pdf-s2} is differentiable with respect to $a$: Now we can calculate the expected value of $Y = \ln(X)$: Using the derivative of a logarithmized function, and the definition of the digamma function. Stack Exchange Network. What seems like related parameterization from Mathematica: https://reference.wolfram.com/language/ref/LogGammaDistribution.html. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. Edit 2: If there's no closed-form expression, is there a better way to approximate this expectation than monte carlo simulation of X? From the mathematica reference: "The LogGammaDistribution[α,β,μ] is equivalent to TransformedDistribution[Exp[x]+μ-1,x ~ GammaDistribution[α,β]]". MathJax reference. To learn more, see our tips on writing great answers. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Generic word for firearms with long barrels. In this case, the probability density function of the gamma distribution is, Multiplying this function with $\mathrm{d}x$, we obtain, Substituting $y = \ln x$, i.e. 2. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. and that a scaling parameter acts additively on the logarithmic expectation of a random variable. The expectation of the $\Gamma$-distribution parametrized as above is $\alpha/\beta$ (like you mentioned), the variance is $\alpha/\beta^2$, hence, the expectation of its square is $(\alpha/\beta)^2+\alpha/\beta^2$. Title of book about humanity seeing their lives X years in the future due to astronomical event. Making statements based on opinion; back them up with references or personal experience. polynomial in $\lambda$ of degree $n-2$. OOP implementation of Rock Paper Scissors game logic in Java. I need help with the step indicated by the red arrow. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … What is the expected number of times S appeared? In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. How do smaller capacitors filter out higher frequencies than larger values? I would like to know if there is a closed form expression for the expectation of log(1+x) when x is a gamma random variable. {\mbox{$_2$F$_2$}(1,1;\,2,2-n;\lambda)}\lambda}{n-1}}+{\frac { How can you trust that there is no backdoor in your hardware? site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. 6. What is this part of an aircraft (looks like a long thick pole sticking out of the back)? If $X$ has the gamma distribution with rate $\lambda$ and shape parameter $n$, you're asking for Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It only takes a minute to sign up. Could you guys recommend a book or lecture notes that is easy to understand about time series? Edit: Note the gamma function inside the log. Edit: Note the gamma function inside the log. The answer is written out here (https://www.physicsforums.com/threads/mean-and-variance-of-loggamma-distribution.779681/), except the final arithmetic. Finally, noting that $1/b$ acts as a scaling parameter on a gamma-distributed random variable. Making statements based on opinion; back them up with references or personal experience. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Gamma distribution is a kind of statistical distributions which is related to the beta distribution. Constructing a Bernoulli random variable for ratio of Bernoulli weights. Proof: Logarithmic expectation of the gamma distribution Index: The Book of Statistical Proofs Probability Distributions Univariate continuous distributions Gamma distribution Logarithmic expectation \left( -1 \right) ^{-n}\pi \Gamma \left( n,-\lambda \right) }{\sin ; in. This distribution arises naturally in which the waiting time between Poisson distributed events are relevant to each other. Why is it easier to carry a person while spinning than not spinning? rev 2020.11.24.38066, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, expectation of log(1+x) if x is a gamma random variable, “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2/4/9 UTC (8:30PM…, Expectation of RVs with Poisson-type decay, Estimate which random variable has highest expectation, expectation of log(x+a) when X follows a beta distribution, Constructing a Bernoulli random variable for ratio of Bernoulli weights, Gaussian expectation of an exponentiated outer product, Bounding integral arising from expectation of a random variable satisfying Bernstein's inequality. How to generate valid BTC address to withdraw coins to. I would like to know if there is a closed form expression for the expectation of log(1+x) when x is a gamma random variable.