Home > Doc > The Partial Distribution:Definition, Properties and Applications in Economy > Properties Analysis

The Partial Distribution:Definition, Properties and Applications in Economy

Properties Analysis

We shall show the basic properties of Partial Distribution by comparing with lognormal and levy.

A.The properties of lognormal distribution Ln( µ, σ2).
1) Stochastic variable is non-negative, and the probability is zero at x=0, namely, p(0)=0.

2) The shape of distribution curve is relevant to parameter σ and variable x.

3) Expectation is

B.The properties of Levy distribution Lα(a,x).
1) Levy distribution is the probability distribution function ζα(a,x) with a characteristic function:

2) Gaussian (α=2, a=σ2/2) and Cauchy (α=1) distribution are the only two levy distributions that can
be inverted and expressed as elementary functions.

3) When 0<α<2, both expectation and variance of levy does not exist.

4) The center of the Levy distribution gets sharper and higher and the tails get fatter as α--->0 and x--->0.

5) the most applications of truncated levy distribution is on both ends at the same time, the discussion
on truncating a single side is not much more.

C.The properties of Partial Distribution P( µ, σ2).

1) Stochastic variable is non-negative, and the probability is non-zero at x=0, namely,

2) The shape of distribution curve is relevant to parameter σ and µ

3) The expectation E(x)=µ+R(x) , i.e. E(x)>µ.

4) The variance D(x)= σ2+E(x)(µ-E(x)), D(x)< σ2.

5) When x--->µ, Partial distribution is sharper than Gaussian and lognormal distribution as µ is less.

6) If µ is big enough, the Partial distribution is approximately near to Gaussian distribution.

The curve shapes of Partial distribution are shown in figure 1.


Prof. Feng Dai

Next: Applications of partial distribution

Summary: Index