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I want to get the distribution of the following quantity:

$\sum_{i=1}^{N} M_i \cdot \overline e $

where:

  • $M_i$ with $i=1,...,N$ are constant values
  • $\overline e$ is a normally distributed random variable, let's say $\overline e \sim N(\mu, \sigma^2)$

My doubt is between these two solutions:

  1. $\sum_{i=1}^{N} M_i \cdot \overline e \:=\: \overline e \, \cdot \sum_{i=1}^{N} M_i \: \sim N(\sum_{i=1}^{N} M_i \cdot \mu, [\sum_{i=1}^{N} M_i]^2 \cdot \sigma^2)$

  2. $\sum_{i=1}^{N} M_i \cdot \overline e \: = \: \sum_{i=1}^{N} (M_i \cdot \overline e) \; \Rightarrow \; (M_i \cdot \overline e) \sim N(M_i\cdot\mu, M_i^2\cdot\sigma^2) \; \\ \Rightarrow \; \sum_{i=1}^{N} M_i \cdot \overline e \: \sim \: N(\sum_{i=1}^{N}M_i\cdot\mu\,,\, \sum_{i=1}^{N}[M_i^2\cdot\sigma^2])$

Which one of the two is correct? Why?

Thank you in advance for your answer!!!

Federico Carrara
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  • This is known universally in mathematics and statistics as a *linear combination* (of random variables). Apply the standard rules for expectation (linearity) and covariance (bilinearity) to obtain the correct result. Searching on these technical terms here will uncover (literally) hundreds of examples you can emulate. I chose only the first hit as the duplicate, but many of the others are worth reading, too. – whuber Oct 06 '21 at 13:12

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