deletions | additions       diff --git a/section_Statistics_The_data_points__.tex b/section_Statistics_The_data_points__.tex  index 7d39786..2b4e016 100644  --- a/section_Statistics_The_data_points__.tex  +++ b/section_Statistics_The_data_points__.tex 
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\section{Statistics}  The data points (quasar pairs) collected for our study offers us the biggest sample space possible, as it was obtained from the biggest spectroscopy survey – the SDSS. Our sample has been created through the use of the criteria in the Liu et al \citet{Liu_2011}  pair catalogue section. The data points are grouped into bins. This study makes use of binomial statistics for data analysis purposes.  The data points – the quasar pairs- follow a binomial distribution where the event of detecting a quasar that is in a quasar pair denotes a success. We have 138 070 Bernoulli trials. Where each trial either results in a success- a radio detection of one of the two AGN in each quasar pair in the FIRST survey ( referring to a quasar pair as defined by the selection criterion employed by Liu et al 2011) or a failure -not detecting a quasar pair. Liu et al \citet{Liu_2011}  detected 1 286 pairs, i.e we obtained 1286 successes. n= 138 070, is the number of trials we conduct or the total number of AGN we have available.The probability of a success is $\frac{1286}{138070} \approx 0.0093$ . We further assume that the trial are independent from each other (under the guidance of our selection criteria). Our variable of interest is the number of successes observed during the n trials. There are about 200 pairs per projected separation bin and the bins are randomly cut to be a similar size. The sample space is cut into 6 bins. The errors in the projected separations of our data points are neglected neglected because the selected bins width ($\sigma_{rp} << 20 kpc$) are much larger than the projected separation uncertainties (as seen in figure Figure  11 and Figure  12).   In Figure 11, we plot the bin averages of radio luminosity against projected separation. Using the binomial theorem each of the y-error bars is calculated using:  $\sigma = \sqrt{np(1-p)}$.   Where n is the number of data points in the particular bin and p is the fraction detected at radio frequencies (1.4 GHz). The project separation measurement uncertainties are assumed to be negligible.  The median radio luminosity of each bin was plotted against the projected separation in figure Figure  12. Since this data is obtained in nature via random sample (and since we have a large sample), we approximate the distribution of the radio luminosities to a normal distribution. The error in the measurement of these luminosities is the conventional standard deviation of a normal distribution/Gaussian. The Gaussian distribution is given by: $f(x) = \frac{1}{2 \pi {\sigma}^2}\exp{(-(\frac{1}{2{\sigma}^2}){(x - \mu)}^2)}$