大家帮帮忙，什么叫矩母函数？

hc3256

<>大家帮帮忙，什么叫矩母函数？如果有相关资料，能不能贴一下，或发我邮箱<a href="http://www.hc3256@163.com" target="_blank" >www.hc3256@163.com</A></P>
<>谢谢了！！！！！！！</P>

panzervi

<>转自mathworld.wolfram.com</P><TABLE cellSpacing=0 cellPadding=0 border=0><TR><TD class=title vAlign=baseline><NOBR>Moment-Generating Function</NOBR></TD></TR><TR><TD vAlign=top><NOBR><img src="http://mathworld.wolfram.com/images/entries/underline.gif"><img src="http://mathworld.wolfram.com/images/entries/underline.gif"></NOBR></TD><TD><img src="http://mathworld.wolfram.com/images/spacer.gif"></TD></TR></TABLE><!-- End Title Tab --><!-- Main Page --><DIV class=entry><>Given a <a href="http://mathworld.wolfram.com/RandomVariable.html" target="_blank" >random variable</A> <I>x</I> and a <a href="http://mathworld.wolfram.com/ProbabilityDistributionFunction.html" target="_blank" >probability distribution function</A> <I></I>(<I>x</I>), if there exists an <I>h</I> &gt; 0 such that <DIV align=right><TABLE width="100%" align=center><TR vAlign=center><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg3532.gif"></TD><TD align=right width=10>(1)</TD></TR></TABLE><BR clear=all></DIV><P>for <NOWRAP><img src="http://mathworld.wolfram.com/mimg3533.gif">,</NOWRAP> where <img src="http://mathworld.wolfram.com/mimg3534.gif"> denotes the <a href="http://mathworld.wolfram.com/ExpectationValue.html" target="_blank" >expectation value</A> of <I>y</I>, then <I>M</I>(<I>t</I>) is called the moment-generating function. <P>For a continuous distribution, <DIV align=center><TABLE cellPadding=0 width="100%" align=center><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3536.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3537.gif"></TD><TD align=right width=10>(2)</TD></TR><TR vAlign=center><TD noWrap align=right> </TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3538.gif"></TD><TD align=right width=10>(3)</TD></TR><TR vAlign=center><TD noWrap align=right> </TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3539.gif"></TD><TD align=right width=10>(4)</TD></TR></TABLE></DIV><BR clear=all>where <img src="http://mathworld.wolfram.com/mimg3540.gif"> is the <I>r</I>th <a href="http://mathworld.wolfram.com/CentralMoment.html" target="_blank" >central moment</A>. <P>For independent <I>X</I> and <I>Y</I>, the moment-generating function satisfies <DIV align=center><TABLE cellPadding=0 width="100%" align=center><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3541.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3542.gif"></TD><TD align=right width=10>(5)</TD></TR><TR vAlign=center><TD noWrap align=right> </TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3543.gif"></TD><TD align=right width=10>(6)</TD></TR><TR vAlign=center><TD noWrap align=right> </TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3544.gif"></TD><TD align=right width=10>(7)</TD></TR></TABLE></DIV><BR clear=all><P>If <I>M</I>(<I>t</I>) is differentiable at zero, then the <I>n</I>th <a href="http://mathworld.wolfram.com/Moment.html" target="_blank" >moments</A> about the <a href="http://mathworld.wolfram.com/Origin.html" target="_blank" >origin</A> are given by <img src="http://mathworld.wolfram.com/mimg3545.gif"> <DIV align=center><TABLE cellPadding=0 width="100%" align=center><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3536.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3546.gif"></TD><TD align=right width=10>(8)</TD></TR><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3547.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3548.gif"></TD><TD align=right width=10>(9)</TD></TR><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3549.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3550.gif"></TD><TD align=right width=10>(10)</TD></TR><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3551.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3552.gif"></TD><TD align=right width=10>(11)</TD></TR></TABLE></DIV><BR clear=all>The <a href="http://mathworld.wolfram.com/Mean.html" target="_blank" >mean</A> and <a href="http://mathworld.wolfram.com/Variance.html" target="_blank" >variance</A> are therefore <DIV align=center><TABLE cellPadding=0 width="100%" align=center><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg1371.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg703.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3553.gif"></TD><TD align=right width=10>(12)</TD></TR><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg1372.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg703.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3554.gif"></TD><TD align=right width=10>(13)</TD></TR></TABLE></DIV><BR clear=all>It is also true that <DIV align=right><TABLE width="100%" align=center><TR vAlign=center><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg3555.gif"></TD><TD align=right width=10>(14)</TD></TR></TABLE><BR clear=all></DIV><P>where <img src="http://mathworld.wolfram.com/mimg3556.gif"> and <img src="http://mathworld.wolfram.com/mimg3557.gif"> is the <I>j</I>th moment about the origin. <P>It is sometimes simpler to work with the <a href="http://mathworld.wolfram.com/Logarithm.html" target="_blank" >logarithm</A> of the moment-generating function, which is also called the <a href="http://mathworld.wolfram.com/Cumulant-GeneratingFunction.html" target="_blank" >cumulant-generating function</A>, and is defined by <DIV align=center><TABLE cellPadding=0 width="100%" align=center><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3558.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg703.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3559.gif"></TD><TD align=right width=10>(15)</TD></TR><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3560.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3561.gif"></TD><TD align=right width=10>(16)</TD></TR><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg3562.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3563.gif"></TD><TD align=right width=10>(17)</TD></TR></TABLE></DIV><BR clear=all>But <NOWRAP><img src="http://mathworld.wolfram.com/mimg3564.gif">,</NOWRAP> so <DIV align=center><TABLE cellPadding=0 width="100%" align=center><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg1371.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3565.gif"></TD><TD align=right width=10>(18)</TD></TR><TR vAlign=center><TD noWrap align=right><img src="http://mathworld.wolfram.com/mimg1372.gif"></TD><TD noWrap align=middle><img src="http://mathworld.wolfram.com/mimg65.gif"></TD><TD noWrap align=left><img src="http://mathworld.wolfram.com/mimg3566.gif"></TD><TD align=right width=10>(19)</TD></TR></TABLE></DIV><BR clear=all><P><P><img src="http://mathworld.wolfram.com/images/entries/see_also.gif"><a href="http://mathworld.wolfram.com/CharacteristicFunction.html" target="_blank" >Characteristic Function</A>, <a href="http://mathworld.wolfram.com/Cumulant.html" target="_blank" >Cumulant</A>, <a href="http://mathworld.wolfram.com/Cumulant-GeneratingFunction.html" target="_blank" >Cumulant-Generating Function</A>, <a href="http://mathworld.wolfram.com/Moment.html" target="_blank" >Moment</A> </P></DIV>

panzervi

呃……贴出来都是代码……运行下可以看……

hc3256

<> <FONT color=#990000><B>panzervi</B></FONT>  谢谢你了!</P>