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