Gaussian distribution determined by first two moments When said that Gaussian distribution is determined by it's mean and variance. How is that different of other distributions? Almost every distribution which I can think of has this property. For example if we know the mean of exponential, Poisson distribution then we know the whole distributions.  
 A: I agree with the comment by @eigenchris for 'well-known' distributions encountered early on in a probability course. However, one does not have
to venture too far into the study of probability distributions to
find examples in which knowing the population mean and variance
does not easily specify the distribution.
It is useful to make a distinction between the parameters of the
distribution of a random variable $X$ and other quantities such as $\mu =E(X), \sigma^2=V(X),$ and $E(X^2),$ (sometimes called 'moments') which in some sense might be said to 'determine' the distribution. Here are a few examples: 
UNIFORM: If $X \sim Unif(\alpha_1, \alpha_2)$, then the endpoints $\alpha_1$ and $\alpha_2$ of the support interval are usually taken
as the parameters of the distribution. However, if specified, the mean 
$\mu = E(X) = (\alpha_1 + \alpha_2)/2$ and 
variance $\sigma^2 = V(X) = (\alpha_2 - \alpha_1)^2/12$ could be
used to find the parameters $\alpha_1$ and $\alpha_2$ in terms
of $\mu$ and $\sigma^2.$
GAMMA:  If $X \sim Gamma(\alpha, \theta$), then $\alpha$ is the
shape parameter and $\theta$ is the scale parameter. Again, if
$\mu = \alpha\theta$ and $\sigma^2 = \alpha\theta^2$ are known,
then we could easily solve to find the parameters $\alpha$ and $\theta$
in terms of $\mu$ and $\sigma^2.$
In both uniform and gamma distributions, it is usually more natural
or intuitive to think in terms of the parameters, even though they
are straightforwardly determined by $\mu$ and $\sigma^2.$
In some other families of distributions, the relationship between
moments $\mu$ and $\sigma^2$ and the more natural parameters is not
expressed so transparently.
BETA. This family of distributions has two parameters $\alpha$ and $\beta.$ These distributions have support $(0, 1)$. Very roughly
speaking $\alpha$ controls the 'shape' of the distribution near $0$ and $\beta$ controls shape near $1$. Here $\mu = \alpha/(\alpha + \beta)$ and $\sigma^2 = 
\frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}.$
It is possible, but not really easy or intuitive to use the moments
to determine the parameters.
WEIBULL. This family of distributions has a shape parameter $\kappa$
and a scale parameter $\lambda.$ It is often used in reliability
theory and economics. Here $\mu = \lambda \Gamma(1 + 1/\kappa),$
where $\Gamma$ is the gamma function;  $\sigma^2$ is expressed
in terms of a somewhat more complex formula involving two
$\Gamma$ functions and the parameters (see Wikipedia). 
In applications of these last two families of distributions, it is
much more natural to think in terms of the parameters than in terms
of the moments, even though numerical methods can be used to
find the parameters if specific values of the moments are given.
