Question: I have been reading Davidson's Econometric theory and methods and I cannot understand one particular paragraph about the distribution of a test statistic.In the paragraph below he says that under the alternative hypothesis if the true $\beta$ is equal to $\beta\_1$ then $\hat b= \beta\_1+\hat\gamma$. Then he defines $z=(\beta\_1+\hat\gamma-\beta\_0)/(\sigma^2/\nu)$ and finds the mean and variance as given by 4.04. Why is he doing that? I understand how he gets the mean and variance but I don't get why is he doing that because at the end we dont get a standard normal variable but a variable with mean $\lambda$ and variance 1.[![enter image description here](https://i.stack.imgur.com/r8wtR.png)](https://i.stack.imgur.com/r8wtR.png)

I have been stuck here for a while so any help would be highly appreciated.

Answer: 
I will try to do this at an intuitive level without being technical. If you want rigorous definitions of any of this I can edit my post. 

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We need a way to mathematically model the consumption choices that an agent faces. To accomplish this, we use $\mathbb{R}^L\_+$ to denote the agent's level of consumption of each of 1,2,..,L goods. 

Vectors tie into this because we want to think about how much of each good an agent chooses when optimizing over her preferences. So we don't just think about how much of good $i$ the agent wants. We want to think about how much of good 1 the agent wants, good $2$ he wants, etc. simultaneously. So, there are tons of different "mixes" of these goods that an agent can afford (that belong to this agent's budget set given some prices and wealth) and one of those "mixes" (vectors) represents how much of each of the L goods an agent chooses to consume when maximizing his or her representative utility. 

Now - Cartesian space. You should actually discuss the Cartesian coordinate system on Euclidean space. We use Cartesian coordinates to denote a point in Euclidean space. We could, for example, instead use polar coordinates to denote points in Euclidean space. 

A metric space is a bit more complicated but you can think of it (this is very simplified) as a space where the distance between any two points is well defined. The distances themselves are called a metric. Having a metric space allows us to use topological methods (for example, the concept of open balls when discussing the local non-satiation of preferences). The most common example of a metric space is three-dimensional Euclidean space.
Of all the types of structures or spaces you mention, vector space $\mathcal{V}$ is the simplest type of structure. Other structures are constructed from this:

1. A real vector space with an [*inner product*](https://en.wikipedia.org/wiki/Inner_product_space) (which allows to define the ortogonality relation) is a [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space) $\mathcal{H}$.
2. An Euclidean space is an [affine space](https://en.wikipedia.org/wiki/Affine_space) $\mathcal{A}$, a set of points, such that for every pair of points $P$ and $Q$ there is a a vector $\overrightarrow{PQ}$ of a finite dimensional Hilber space. In an Euclidean space you can define, distances, angles, points, etc. so you have a geometrical space.
3. A set of points with a distance is a metric space, Euclidean space is a metric space, but not every metric space is an Euclidean space.
4. A [commodity space](https://en.wikipedia.org/wiki/Debreu_theorems) is a set of points, where you can define a total order relation, which is continuous and has additional properties (some convex sets of the Euclidean spaces can become a commodity space, if you define some additional structure on them).
5. A Cartesian space, is just a set that can be expressed as [Cartesian product](https://en.wikipedia.org/wiki/Cartesian_product) of sets. Again, mutidimensional Euclidean space and many commodity spaces are particular instances of Cartesian spaces.