d*******n
发帖数: 216 | 1 If two distributions have the same expectation and variance, do they have
the same cumulative distribution function? My hunch is not, but cannot
think of an example.( I know the density functions can be different, for
example, at finitely countable points, but I want to know about more general
cases.)
I'd appreciate any help. Thanks! | C******t
发帖数: 72 | 2 A simple example: x ~ N (1/2,1/12), Y~ Uniform (0,1). Both of them have the expectation 1/2, and variance 1/12, but they are different distribution.
In more general scienario, there is a theorem: Let Fx(X) and Fy(Y)be two
cdfs all of whose moments existl If X and Y have bounded support, the Fx(u)=
Fy(u) for all u if and only if Ex^r=Ex^r for all intergers r=0,1,2,....;or
if the moment generating unctions exist and Mx(t)=My(t) for all t in some
neighborhood of 0, the Fx(u)=Fy(u) for all u. | d*******n
发帖数: 216 | 3 Thanks a lot for the help. Learned this theorem a few years ago in a
probability class, but somehow had completely forgot about it. |
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