NO.PZ2020010303000008
问题如下:
How are the mean and variance of a mixture of normal random variables related to the mean and variance of the components of the mixture?
解释:
The mean is the weighted average of the means of the components. The variance is more complicated. The second non-central moment, of the mixture is the weighted average of the second non-central moments of the components. The variance is then , which depends the first and second moments of the mixture.
老师 我先想确认一下mixture distribution的定义里是不是有要求weight必须是random variable而非常数?如果是的话,那题干中mixture of normal distribution和linear combination of normal distribution不是就不一样了吗,这个时候expectation还能直接根据weight这个random variable的每种取值对应的概率直接加权平均吗,weight已经不是常数了,那答案中说的期望是加权平均具体是怎么算的呢。或者说老师能否举一个有3个component distribution的 mixture的例子呢,这个时候W就不能是Bernoulli而是其他分布了,那求期望的时候不是要按定义计算,一定能够推导到加权平均的形式吗?我不要太理解为什么答案中关于期望和方差的部分是成立的,感觉题干问得是一种比较general的情况。谢谢老师!