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Milds # Lomax distribution

Parameters Probability density function Cumulative distribution function $\alpha >0$ shape (real) $\lambda >0$ scale (real) $x\geq 0$ ${\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)}$ $1-\left[{1+{x \over \lambda }}\right]^{-\alpha }$ ${\lambda \over {\alpha -1}}{\text{ for }}\alpha >1$ Otherwise undefined $\lambda ({\sqrt[{\alpha }]{2}}-1)$ 0 ${{\lambda ^{2}\alpha } \over {(\alpha -1)^{2}(\alpha -2)}}{\text{ for }}\alpha >2$ $\infty {\text{ for }}1<\alpha \leq 2$ Otherwise undefined ${\frac {2(1+\alpha )}{\alpha -3}}\,{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3\,$ ${\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4\,$ The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.

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#### Transcription

Innovation is the driving force of a knowledge-based economy. While innovation has many sources, science and technology are its fundamental elements. Now more than ever, innovation is the essential ingredient for continued economic growth and well-being. Currently, North Carolina's overall innovation ranking among US states is just average. When it comes to innovation from the middle of the pack, North Carolina has opportunity for tremendous improvement. Our academic institutions excel at research and development and produce well-trained graduates. However, corporate research activity lags behind other states. We also have fewer entrepreneurs and provide lower levels of venture capital to support new business formation. A study using SAS Visual Analytics software found that states in the top 25% for output and compensation are strongly linked to high levels of three key indicators: post-secondary educational attainment, the proportion of workers in high-tech industries, and the proportion of workers in science and engineering occupations across the economy. On these three measures, North Carolina is mired well below states in the top 25%, trailing even the US average. More concerning, recent trend lines offer no sign that North Carolina will reach the top 25% in the foreseeable future. To move into the top tier of states, North Carolina must raise education and training levels so that more people are qualified to work in technical jobs, deepen the technology level of all existing companies, and start and growth new high-tech companies. For more North Carolinians to benefit, this must be done across the state. Innovation-creating activity remains concentrated geographically, though this is changing. Awards by the federal government's primary technology development and commercialization program for small businesses show a marked increase outward from the triangle area, a reflection of deepening research capacities in other parts of the state. Such investment in early stage research and development is a precondition for generating new technologies, companies, and jobs. We see increased education levels across the state. In recent decades, the state's percentage of residents with a bachelor's degree or higher rose considerably. Despite these advances, many of our counties remained below the education levels of states in the top 25% for output and compensation. We also see a wider distribution of high-tech employment across North Carolina. In 1990, 95% of the state's high-tech jobs were concentrated in just 13 counties, compared to 22 counties today. Our current trajectory is clear. North Carolina will remain average for innovation unless we deepen existing innovation pockets and broaden emerging capacities for innovation in other areas of the state. What we choose to do today will chart a new course for decades to come.

## Characterization

### Probability density function

The probability density function (pdf) for the Lomax distribution is given by

$p(x)={\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)},\qquad x\geq 0,$ with shape parameter $\alpha >0$ and scale parameter $\lambda >0$ . The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is:

$p(x)={{\alpha \lambda ^{\alpha }} \over {(x+\lambda )^{\alpha +1}}}$ .

### Non-central moments

The $\nu$ th non-central moment $E[X^{\nu }]$ exists only if the shape parameter $\alpha$ strictly exceeds $\nu$ , when the moment has the value

$E(X^{\nu })={\frac {\lambda ^{\nu }\Gamma (\alpha -\nu )\Gamma (1+\nu )}{\Gamma (\alpha )}}$ ## Related distributions

### Relation to the Pareto distribution

The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:

${\text{If }}Y\sim {\mbox{Pareto}}(x_{m}=\lambda ,\alpha ),{\text{ then }}Y-x_{m}\sim {\mbox{Lomax}}(\alpha ,\lambda ).$ The Lomax distribution is a Pareto Type II distribution with xm=λ and μ=0:

${\text{If }}X\sim {\mbox{Lomax}}(\alpha ,\lambda ){\text{ then }}X\sim {\text{P(II)}}(x_{m}=\lambda ,\alpha ,\mu =0).$ ### Relation to the generalized Pareto distribution

The Lomax distribution is a special case of the generalized Pareto distribution. Specifically:

$\mu =0,~\xi ={1 \over \alpha },~\sigma ={\lambda \over \alpha }.$ ### Relation to the beta prime distribution

The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then ${\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )$ .

### Relation to the F distribution

The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density $f(x)={\frac {1}{(1+x)^{2}}}$ , the same distribution as an F(2,2) distribution. This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions.

### Relation to the q-exponential distribution

The Lomax distribution is a special case of the q-exponential distribution. The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by:

$\alpha ={{2-q} \over {q-1}},~\lambda ={1 \over \lambda _{q}(q-1)}.$ ### Relation to the (log-) logistic distribution

The logarithm of a Lomax(shape=1.0, scale=λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0. This implies that a Lomax(shape=1.0, scale=λ)-distribution equals a log-logistic distribution with shape β=1.0 and scale α=log(λ).

### Gamma-exponential mixture connection

The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution. If λ|k,θ ~ Gamma(shape=k, scale=θ) and X|λ ~ Exponential(rate=λ) then the marginal distribution of X|k,θ is Lomax(shape=k, scale=1/θ).