FAQ/gamma - CBU statistics Wiki

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# How do I produce random variables which follow a negatively skewed distribution?

Most distributions such as the exponential and log-Normal distributions are positively skewed with the mode of the distribution occurring for lower values.

The Gamma distribution [http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_gamma_distri.htm: may be produced by summing exponential distributions.] There are two parameters, n and $$\lambda$$. n is the number of summed exponentials and $$\lambda$$ is the exponential rate. When $$\lambda$$ is very small compared to n a negative skewed version of the gamma distribution results with no upper limit.

The below produces two exponential random variables (n=2) with a very small $$\lambda$$ (=1/10000) in SPSS and sums them. One resulting simulated data set produced using this macro had a skew of -1.03.

define !gamma ( !pos !tokens(1)
/!pos !tokens(1)).
!do !i=!1 !to !2 !by 1.
compute !concat(a,!i)=-(10000)*ln(rv.uniform(0,1)*10000).
!doend.
!enddefine.

!gamma 1 2.
exe.

compute sum=0.
exe.

compute sum=-(a1+a2).
exe.