Question

Asked Feb 22, 2019

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Recall that we say that a random variable X is in the vector space L^2 if it has finite second moment, EX^2 In this problem we will understand a bit better the geometry of the vector space L^2

(1) Show that ||X||_2 = √EX^2 is a norm

Step 1

**Properties of a norm:**

In order to prove that ||*X*||_{2} = √*E*(*X*^{2}) is a norm, it must satisfy 3 conditions:

- Positivity,
- Homogeneity,
- Sub-additivity.

Step 2

**Proof of positivity:**

To prove: ||*x*||_{2} ≥ 0 for all *x *ϵ *X* and *x* = 0 if and only if *x* = 0.

Proof:

Step 3

**Proof of homogeneity:**

To prove: For all *x *ϵ *X*, ||*cx*||2 = |*c*|.||*x*||2 for all real *c*...

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