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As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer. This is how the symbols are used at least in Statistics/Econometrics.

As regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the parameters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function. We do not note down the moments, which usually are a function of, but not equal to, these parameters.

So for a Uniform that ranges in $[a,b]$ we write $U(a,b)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean is $k\theta$ and the variance $k\theta^2$. Etc.

In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$ (which conforms with the general notational rule), while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$ (which falls to the temptation of providing the moments, by treating $\sigma^2$ as the base parameter and not as the square of it).

As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer. This is how the symbols are used at least in Statistics/Econometrics.

As regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the parameters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function. We do not note down the moments, which usually are a function of, but not equal to, these parameters.

So for a Uniform that ranges in $[a,b]$ we write $U(a,b)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean is $k\theta$ and the variance $k\theta^2$. Etc.

In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$ (which conforms with the general notational rule), while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$ (which falls to the temptation of providing the moments, by treating $\sigma^2$ as the base parameter and not as the square of it).

As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer. This is how the symbols are used at least in Statistis/Econometrics.

As regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the paramaters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function. We do not note down the moments, which usually are a function of, but not equal to, these parameters.

So for a Uniform that ranges in $[a,b]$ we write $U(a,b)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean is $k\theta$ and the variance $k\theta^2$. Etc.

In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$ (which conforms with the general notational rule), while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$ (which falls to the temptation of providing the moments, by treating $\sigma^2$ as the base parameter and not as the square of it).

As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer. This is how the symbols are used at least in Statistics/Econometrics.

As regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the parameters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function. We do not note down the moments, which usually are a function of, but not equal to, these parameters.

So for a Uniform that ranges in $[a,b]$ we write $U(a,b)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean is $k\theta$ and the variance $k\theta^2$. Etc.

In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$ (which conforms with the general notational rule), while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$ (which falls to the temptation of providing the moments, by treating $\sigma^2$ as the base parameter and not as the square of it).

As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer. This is how the symbols are used at least in Statistis/Econometrics.

As regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the paramaters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function. We do not note down the moments, which usually are a function of, but not equal to, these parameters.

So for a Uniform that ranges in $[a,b]$ we write $U(a,b)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean is $k\theta$ and the variance $k\theta^2$. Etc.

In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$, while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$.

As regards the use of symbols $\sim$ ("follows", "is distributed according to "), and $\approx$ ("equals approximately"), see this answer. This is how the symbols are used at least in Statistis/Econometrics.

As regards the notational conventions for a distribution, the normal is a borderline case: we usually write the defining parameters of a distribution alongside its symbol, the paramaters that will permit one to write correctly its Cumulative distribution function and its probability density/mass function. We do not note down the moments, which usually are a function of, but not equal to, these parameters.

So for a Uniform that ranges in $[a,b]$ we write $U(a,b)$. The mean of the distribution is $(a+b)/2$ while the variance is $(b-a)^2/12$. For a Gamma (shape-scale parametrization), we write $G(k,\theta)$. The mean is $k\theta$ and the variance $k\theta^2$. Etc.

In the case of the normal distribution, the parameter $\mu$ happens to also be the mean of the distribution, while the parameter $\sigma$ happens to be the square root of the variance. It is my (possibly mistaken) impression that in Engineering circles one sees more often $N(\mu, \sigma)$ (which conforms with the general notational rule), while in Econometrics circles almost always one sees $N(\mu, \sigma^2)$ (which falls to the temptation of providing the moments, by treating $\sigma^2$ as the base parameter and not as the square of it).