Probability density function
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Parameters | (shape parameter) | ||
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Support | |||
Mean | |||
Variance |
The Schulz–Zimm distribution is a special case of the gamma distribution. It is widely used to model the polydispersity of polymers. In this context it has been introduced in 1939 by Günter Victor Schulz [1] and in 1948 by Bruno H. Zimm. [2]
This distribution has only a shape parameter k, the scale being fixed at θ=1/k. Accordingly, the probability density function is
When applied to polymers, the variable x is the relative mass or chain length . Accordingly, the mass distribution is just a gamma distribution with scale parameter . This explains why the Schulz–Zimm distribution is unheard of outside its conventional application domain.
The distribution has mean 1 and variance 1/k. The polymer dispersity is .
For large k the Schulz–Zimm distribution approaches a Gaussian distribution. In algorithms where one needs to draw samples , the Schulz–Zimm distribution is to be preferred over a Gaussian because the latter requires an arbitrary cut-off to prevent negative x.
Probability density function
| |||
Parameters | (shape parameter) | ||
---|---|---|---|
Support | |||
Mean | |||
Variance |
The Schulz–Zimm distribution is a special case of the gamma distribution. It is widely used to model the polydispersity of polymers. In this context it has been introduced in 1939 by Günter Victor Schulz [1] and in 1948 by Bruno H. Zimm. [2]
This distribution has only a shape parameter k, the scale being fixed at θ=1/k. Accordingly, the probability density function is
When applied to polymers, the variable x is the relative mass or chain length . Accordingly, the mass distribution is just a gamma distribution with scale parameter . This explains why the Schulz–Zimm distribution is unheard of outside its conventional application domain.
The distribution has mean 1 and variance 1/k. The polymer dispersity is .
For large k the Schulz–Zimm distribution approaches a Gaussian distribution. In algorithms where one needs to draw samples , the Schulz–Zimm distribution is to be preferred over a Gaussian because the latter requires an arbitrary cut-off to prevent negative x.