A generalized exponential (GEX) distribution described and plotted in a previous blog post, is often used to empirically approximate a wide variety of distribution. Unlike the lognormal distribution, which has two parameters, the GEX distribution has 3 parameters, which understandably permits greater flexibility.
The weight- and number- distributions are given by
\[ W(M) = \frac{m t^{\frac{k+1}{m}}}{\Gamma \left(\frac{k+1}{m}\right)} M^k e^{-t M^m}\]
\[ N(M) = \frac{m t^{k/m}}{\Gamma \left(\frac{k}{m}\right)} M^{k-1}e^{-t M^m} \]
The weight- and number-average molecular weights are:
\[M_w = \int_0^\infty M W(M) dM = t^{-1/m} \frac{\Gamma\left( \frac{2 + k}{m} \right)}{\Gamma\left( \frac{1 + k}{m} \right)}\]
The weight- and number- distributions are given by
\[ W(M) = \frac{m t^{\frac{k+1}{m}}}{\Gamma \left(\frac{k+1}{m}\right)} M^k e^{-t M^m}\]
\[ N(M) = \frac{m t^{k/m}}{\Gamma \left(\frac{k}{m}\right)} M^{k-1}e^{-t M^m} \]
The weight- and number-average molecular weights are:
\[M_w = \int_0^\infty M W(M) dM = t^{-1/m} \frac{\Gamma\left( \frac{2 + k}{m} \right)}{\Gamma\left( \frac{1 + k}{m} \right)}\]
\[M_n = \int_0^\infty M N(M) dM = t^{-1/m} \frac{\Gamma\left( \frac{1 + k}{m} \right)}{\Gamma\left( \frac{k}{m} \right)} \]
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