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Dhanya Thattil c477970cf0 manual updated
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<H3><A NAME="SECTION00625800000000000000">
Numerical comparison of averages</A>
</H3>
<P>
In the next table numerical results are displayed. An exact random-Poisson generator has been used to generate Poisson deviates of given average value
<IMG
WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img181.png"
ALT="$ \lambda$">
, with <!-- MATH
$\lambda=1,10,100,\ldots,1000000$
-->
<IMG
WIDTH="188" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img195.png"
ALT="$ \lambda=1,10,100,\ldots,1000000$">
. For each value <IMG
WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img181.png"
ALT="$ \lambda$">
<IMG
WIDTH="63" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img196.png"
ALT="$ N=10^8$">
deviates have been generated. Then averages have been taken for each value <IMG
WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img181.png"
ALT="$ \lambda$">
and compared with the true value.
For each value <IMG
WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img181.png"
ALT="$ \lambda$">
- in order to have a scale for comparison -
we evaluate the expected absolute s.d. of averages as <!-- MATH
$\xi_\lambda=\sqrt{\lambda/N}$
-->
<IMG
WIDTH="88" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img197.png"
ALT="$ \xi_\lambda=\sqrt{\lambda/N}$">
, and the relative s.d. of averages as <!-- MATH
$\epsilon_\lambda=\sqrt{\lambda/N}/\lambda=1/\sqrt{N\lambda}$
-->
<IMG
WIDTH="179" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img198.png"
ALT="$ \epsilon_\lambda=\sqrt{\lambda/N}/\lambda=1/\sqrt{N\lambda}$">
. Then - for each averaging method - we evaluate the error <IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
(average minus <IMG
WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img181.png"
ALT="$ \lambda$">
),
the relative error <!-- MATH
$e_\lambda=E_\lambda/\lambda$
-->
<IMG
WIDTH="79" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img200.png"
ALT="$ e_\lambda=E_\lambda/\lambda$">
, and finally the comparison criterion <!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
(bold). The comparison criterion is expected to be close to 1 in absolute value. Values much larger than one mean that we are introducing a systematic error.
<P>
<P>
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<FONT SIZE="-1"> </FONT></FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT" COLSPAN=4><FONT SIZE="-1"> <IMG
WIDTH="31" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img202.png"
ALT="$ \lambda =$">
1. ; <!-- MATH
$\xi_\lambda =$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img203.png"
ALT="$ \xi_\lambda = $">
0.0001 ; <!-- MATH
$\epsilon_\lambda$
-->
<IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img204.png"
ALT="$ \epsilon_\lambda$">
= 0.0001</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img205.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle^*$
-->
<IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img163.png"
ALT="$ \langle x\rangle^*$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img192.png"
ALT="$ \langle x \rangle$">
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Averages </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1.303772380383934 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.9999155361216990 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1.581941754994651 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.9999283300000000 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3037723803839338 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.8446387830096658E-04 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.5819417549946508 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.7166999999996815E-04</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img207.png"
ALT="$ e_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3037723803839338 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.8446387830096658E-04 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.5819417549946508 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.7166999999996815E-04</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 3037.723803839338 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -0.8446387830096658 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 5819.417549946508 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -0.7166999999996815 </B> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT" COLSPAN=4><FONT SIZE="-1"> <IMG
WIDTH="31" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img202.png"
ALT="$ \lambda =$">
10.000000000000002 ; <!-- MATH
$\xi_\lambda =$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img203.png"
ALT="$ \xi_\lambda = $">
0.00031622776601683794 ; <!-- MATH
$\epsilon_\lambda$
-->
<IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img204.png"
ALT="$ \epsilon_\lambda$">
= 0.00003162277660168379</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img205.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle^*$
-->
<IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img163.png"
ALT="$ \langle x\rangle^*$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img192.png"
ALT="$ \langle x \rangle$">
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Averages </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 8.848248847530357 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 10.00025732384808 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 10.00052232372917 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 10.00006800000000 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -1.151751152469645 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.2573238480785278E-03 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.5223237291644978E-03 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.6799999999884676E-04</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img207.png"
ALT="$ e_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.1151751152469645 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.2573238480785278E-04 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.5223237291644977E-04 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.6799999999884675E-05</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -3642.156939527943 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.8137294562072904 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 1.651732660112730 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.2150348808878029 </B> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT" COLSPAN=4><FONT SIZE="-1"> <IMG
WIDTH="31" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img202.png"
ALT="$ \lambda =$">
100.00000000000004 ; <!-- MATH
$\xi_\lambda =$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img203.png"
ALT="$ \xi_\lambda = $">
0.0010000000000000002 ; <!-- MATH
$\epsilon_\lambda$
-->
<IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img204.png"
ALT="$ \epsilon_\lambda$">
= 0.000009999999999999997</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img205.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle^*$
-->
<IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img163.png"
ALT="$ \langle x\rangle^*$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img192.png"
ALT="$ \langle x \rangle$">
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Averages </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 98.98978896904168 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 100.0001037814804 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 100.0002153600000 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 100.0002153600000 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -1.010211030958359 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1037814803765968E-03 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.2153599999559219E-03 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.2153599999559219E-03</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img207.png"
ALT="$ e_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.1010211030958359E-01 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1037814803765968E-05 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.2153599999559218E-05 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.2153599999559218E-05</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -1010.211030958359 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.1037814803765968 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.2153599999559219 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.2153599999559219 </B> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT" COLSPAN=4><FONT SIZE="-1"> <IMG
WIDTH="31" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img202.png"
ALT="$ \lambda =$">
1000.0000000000007 ; <!-- MATH
$\xi_\lambda =$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img203.png"
ALT="$ \xi_\lambda = $">
0.0031622776601683803 ; <!-- MATH
$\epsilon_\lambda$
-->
<IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img204.png"
ALT="$ \epsilon_\lambda$">
= 0.000003162277660168378</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img205.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle^*$
-->
<IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img163.png"
ALT="$ \langle x\rangle^*$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img192.png"
ALT="$ \langle x \rangle$">
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Averages </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 999.0029754507847 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1000.003978305674 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1000.003836760000 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1000.003836760000 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.9970245492160075 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3978305673513205E-02 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3836759999330752E-02 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3836759999330752E-02</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img207.png"
ALT="$ e_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.9970245492160069E-03 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3978305673513202E-05 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3836759999330750E-05 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.3836759999330750E-05</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -315.2868458625229 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 1.258050715667192 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 1.213290043331128 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 1.213290043331128 </B> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT" COLSPAN=4><FONT SIZE="-1"> <IMG
WIDTH="31" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img202.png"
ALT="$ \lambda =$">
10000.00000000001 ; <!-- MATH
$\xi_\lambda =$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img203.png"
ALT="$ \xi_\lambda = $">
0.010000000000000005 ; <!-- MATH
$\epsilon_\lambda$
-->
<IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img204.png"
ALT="$ \epsilon_\lambda$">
= 9.999999999999995E-7</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img205.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle^*$
-->
<IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img163.png"
ALT="$ \langle x\rangle^*$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img192.png"
ALT="$ \langle x \rangle$">
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Averages </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 9998.995728116572 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 9999.995828163173 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 9999.995919900000 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 9999.995919900000 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -1.004271883437468 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.4171836835666909E-02 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">-0.4080100008650334E-02 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">-0.4080100008650334E-02</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img207.png"
ALT="$ e_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.1004271883437467E-03 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.4171836835666905E-06 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.4080100008650330E-06 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">-0.4080100008650330E-06</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -100.4271883437468 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -0.4171836835666907 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -0.4080100008650331 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -0.4080100008650331 </B> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT" COLSPAN=4><FONT SIZE="-1"> <IMG
WIDTH="31" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img202.png"
ALT="$ \lambda =$">
100000.0000000002 ; <!-- MATH
$\xi_\lambda =$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img203.png"
ALT="$ \xi_\lambda = $">
0.031622776601683826 ; <!-- MATH
$\epsilon_\lambda$
-->
<IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img204.png"
ALT="$ \epsilon_\lambda$">
= 3.162277660168376E-7</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img205.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle^*$
-->
<IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img163.png"
ALT="$ \langle x\rangle^*$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img192.png"
ALT="$ \langle x \rangle$">
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Averages </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 99999.01275394148 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 100000.0127639189 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 100000.0125627100 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 100000.0125627100 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.9872460587212117 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1276391866849735E-01 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1256270980229601E-01 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1256270980229601E-01</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img207.png"
ALT="$ e_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.9872460587212097E-05 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1276391866849733E-06 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1256270980229599E-06 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1256270980229599E-06</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -31.21946156583365 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.4036305486159527 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.3972677655897895 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 0.3972677655897895 </B> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD></TD>
<TD></TD>
<TD></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT" COLSPAN=4><FONT SIZE="-1"> <IMG
WIDTH="31" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img202.png"
ALT="$ \lambda =$">
1000000.0000000013 ; <!-- MATH
$\xi_\lambda =$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img203.png"
ALT="$ \xi_\lambda = $">
0.10000000000000006 ; <!-- MATH
$\epsilon_\lambda$
-->
<IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img204.png"
ALT="$ \epsilon_\lambda$">
= 9.999999999999993E-8</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img205.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
${\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle^*$
-->
<IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img163.png"
ALT="$ \langle x\rangle^*$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <!-- MATH
$\langle x \rangle$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img192.png"
ALT="$ \langle x \rangle$">
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Averages </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 999999.1188353101 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1000000.118835812 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1000000.118809340 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 1000000.118809340 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$ E_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.8811646911781281 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1188358106883243 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1188093387754634 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1188093387754634 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<IMG
WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img207.png"
ALT="$ e_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> -0.8811646911781270E-06 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1188358106883241E-06 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1188093387754633E-06 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 0.1188093387754633E-06</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
<!-- MATH
$e_\lambda/\epsilon_\lambda$
-->
<IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img201.png"
ALT="$ e_\lambda/\epsilon_\lambda$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> -8.811646911781276 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 1.188358106883242 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 1.188093387754633 </B></FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"><B> 1.188093387754633</B> </FONT></TD>
</TR>
</TABLE>
<P>
<P>
As it is visible from the table:
<DL COMPACT>
<DT>1.&nbsp;&nbsp;&nbsp;&nbsp;</DT>
<DD><!-- MATH
${\langle x \rangle_{\!\mathrm{w(1)}}}\ :$
-->
<IMG
WIDTH="64" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img208.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(1)}}}\ :$">
the weighted average using straight Poisson statistics is consistenty bad at all values of <IMG
WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img181.png"
ALT="$ \lambda$">
, that means at all counting levels;
</DD>
<DT>2.&nbsp;&nbsp;&nbsp;&nbsp;</DT>
<DD><!-- MATH
${\langle x \rangle^*}\$
-->
<IMG
WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img209.png"
ALT="$ {\langle x \rangle^*}\ $">
: the normal average skipping zero count data is bad for <!-- MATH
$\lambda<100$
-->
<IMG
WIDTH="59" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img210.png"
ALT="$ \lambda&lt;100$">
, that means at low counting levels (of course at higher counting levels zeroes are not happening);
</DD>
<DT>3.&nbsp;&nbsp;&nbsp;&nbsp;</DT>
<DD><!-- MATH
${\langle x \rangle}\ \text{and}\ {\langle x \rangle_{\!\mathrm{w(2)}}}$
-->
<IMG
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img211.png"
ALT="$ {\langle x \rangle}\ $">&nbsp; &nbsp;and<IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img206.png"
ALT="$ {\langle x \rangle_{\!\mathrm{w(2)}}}$">
: the normal average including zero count data and the Mighell-Poisson weighted average
are consistently and equivalently good at all counting levels.
</DD>
</DL>
Therefore there is no bias when using the Mighell-Poisson weighted method to average data w.r.t. the usual average. The former, however,
has already accomplished the passage to normal statistics, therefore all operations on data that are not simple averaging can be done in
the framework of normal statistics, where everything is known and clear. In the next section, on the opposite, it is shown that
even simple operations as scaling data lead to the necessity of abandoning Poisson statistics in order to estimate correctly the standard deviations.
<P>
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<ADDRESS>
Thattil Dhanya
2018-09-28
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