# Questions tagged [exponential-family]

A set of distributions (eg, normal, $\chi^2$, Poisson, etc) that share a specific form. Many of the distributions in the exponential family are standard, workhorse distributions in statistics, w/ convenient statistical properties.

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### Convexity of negative log-likelihood of exponential family distribution [closed]

Let $p(y; \theta^Tx) = b(y) \space \exp\big(\theta^T x y - a(\theta^T x)\big)$, where $x$ and $\theta$ are $d$-dimensional vectors and $y$ a scalar. If I'm not mistaken, the negative log-likelihood ...
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### Distribution of the sample covariance of a multivariate exponential family

I am wondering if there is a known form for the distribution of the sample covariance matrix of a random variable that follows a multivariate exponential family distribution. I guess it would be a ...
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### Exponential family expression for bivariate Gaussian

Consider a bivariate Gaussian distribution with the following density: $$f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2)}$$ How can we write it in the exponential ...
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### Distribution of the exponential of a Gamma distributed random variable

I have a random variable $X$ that follows a Gamma distribution. $$X \sim \text{Gamma}[\alpha, \beta]$$ I want to know the distribution of $Y$, i.e., $$Y = a - \exp\left(-b X\right)$$
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### Why is $\theta\mapsto p_{\theta}$ one-to-one $\iff$ $(n+1)$ functions $\{F_1,\ldots,F_n,1\}$ are linearly independent?

If an n-dimensional model $S=\{p_{\theta}| \theta\in\Theta\}$ can be expressed in terms of the functions $\{C,F_1,\ldots,F_n\}$ on a sample space $X$ and a function $\psi\in \Theta$(parameter space) ...
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### Is it possible to derive the probit model by writing it in exponential family form?

We always use latent variable approach to derive probit model, is there any way to derive probit model from the exponential family form (by using link function)? Also, does logit-normal distribution ...
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