# Properties of Expectation

I am currently reading a book "A first course in Linear model Theory". A linear regression model is defined as $$y = X\beta+\epsilon$$ where $$y = (Y_i, ... , Y_N )$$ is an N-dimensional vector of observed responses,$$\beta = ( \beta_0,\beta_1,...\beta_n)$$'is a(k+1)-dimensional vector of unknown parameters,X is an $$N*(k+1)$$ matrix of rank r of known predictors, and $$\epsilon= (\epsilon_1,\epsilon_2,...,\epsilon_n)$$ is N-dimensional random vector of unobserved errors. Suppose,

$$E(\epsilon)=0$$ and $$cov(\epsilon)=\delta ^2 I_N$$ (1)

The book then writes :

Using (1) and the properties of the expectation and covariance operators:

$$E(y)=E(X\beta+\epsilon)=X\beta+E(\epsilon)=X\beta$$ (2)

$$Cov(y)=Cov(X\beta+\epsilon)=Cov(\epsilon)=\delta ^2I_N$$ (3)

I would appreciate so much a proof on how (1) and properties of expectation and covariance lead to equation (2) and (3) conclusions above

When $$X$$ and $$\beta$$ are assumed to be fixed, which is probably the case mentioned in the book, $$X\beta$$ term will be a constant inside the expressions. So, $$\operatorname{var}(\epsilon+c)=\operatorname{\epsilon}, \ \ \ E[\epsilon+c]=E[\epsilon]+c$$ Here, $$c=X\beta$$.
In a fully bayesian model, $$X$$ and $$\beta$$ could have been random as well. In that case, these expressions would have been $$\operatorname{var}(y|X,\beta)$$ and $$E[y|X,\beta]$$.
• ${ x_1,...x_n}$ are random variables. can this be considered fixed. what disturbs me is $E(X\beta )=X\beta$ Commented Feb 17, 2021 at 12:24
• No, if they're assumed to be RVs in this analysis, the correct expression would be $E[X\beta|X]=X\beta$. Either they're assumed fixed in this particular analysis (not referring to another section of the book where $X$ can be assumed as random), or the expectation and variance notation is a bit slack. Commented Feb 17, 2021 at 12:28