Covariance of Sums of Random Variables/Lemma

Theorem
Let $n$ be a strictly positive integer.

Let $\sequence {X_i}_{1 \le i \le n}$ be a sequence of random variables.

Let $Y$ be a random variable.

Then:


 * $\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$

Proof
Proof by induction:

For all $n \in \N$, let $\map P n$ be the proposition:


 * $\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$

Basis for the Induction
We have that:


 * $\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \cov {X_1, Y} = \sum_{i \mathop = 1}^1 \cov {X_i, Y}$

We therefore have that $\map P 1$ is true.

This is our base case.

Induction Hypothesis
Suppose that $\map P n$ is true for some fixed $n \in \N$.

That is:


 * $\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$

We aim to show that it logically follows that $\map P {n + 1}$ is true.

That is:


 * $\displaystyle \cov {\sum_{i \mathop = 1}^{n + 1} X_i, Y} = \sum_{i \mathop = 1}^{n + 1} \cov {X_i, Y}$

Induction Step
This is our induction step:

We have:

Hence the result by induction.