Unit 2 Expected Values, Variance, etc.

2.1 Notation and Some Definitions

In time series, we denote the response variable as $$X_t$$, and clearly the explanatory variable as $$t$$. To denote a specific point in time, let us say $$t = 9$$, we would say $$X_9$$.

The collection of all $$X_n$$ is a stochastic process known as a time series.

2.1.1 Realizations and Ensembles

A realization is a particular instance of a time series. It is one of the infinitely many possible time series that could have been observed. As a time series is a stochastic process, each one could be entirely different. The realization Is the one that was actually observed.

Sometimes a realization is the only realization possible. We can theorize what an infinite population of those realizations, but it is often the case that there is only one.

An ensemble is the theoretical totality of all possible realizations. It is the ensemble of realizations. We can think of that as the population of realizations. When we are thinking of means and variances, we think of them in terms of the ensemble, even though we may only be able to look at a single realization.

2.2 Population Means and Variances

It is important to note that when dealing with an ensemble of realizations, each time point is allowed to have its own mean and variance.

2.2.1 The Mean of $$X_t$$

$$\mu_t$$ is the mean of all possible realizations of $$X_t$$ for a fixed t.

2.2.2 Variance of $$X_t$$

$${\sigma_t}^{2}$$ is the variance of all possible realizations of $$X_t$$ for a fixed t

2.3 Estimating Mean and Variance for Multiple Realizations

Let us say we have $$n$$ realizations of $$X_t$$ at time $$t$$. The average can be then written as

$$\hat{\mu_t}=\frac{1}{n}\sum_{i=1}^{n}X_{t_n}$$

Variance is calculated the same way. We are making sample means and sample variances. If we assume the population is normally distributed, this allows us to take confidence intervals on the mean of our set of realizations.

However, We have a problem: What to do when we only have one realization?

2.4 Expected Value

In short, the expected value $$(E[X])$$ of a random variable (RV) denoted by $$X$$ is the mean or more intuitively the long run average of the event that variable represents.

2.4.1 Discrete RVs

A discrete RV is an RV in which $$X$$ cannot take on any value, it has specific values it can exist at and that is it.

The formula for EV of a discrete RV is as follows: $E\left[X\right] = \sum X P\left(X\right) = \mu$

2.4.2 Continuous RVs

A continuous RV has instead of discrete values a probability distribution/density function, $$f(x)$$.

The formula for EV of a continuous RV is as follows: $E\left[X\right] = \int_{a}^{b} xf\left(x\right)dx = \mu$

Note that is directly analagous to the discrete RV, and that a and b can span to $$\pm\infty$$

Important Note $E\left[X\right] = \mu$ That is the expected value of a RV is equivalent to the population mean

2.4.3 Some Rules

Let a and b be constants

$E\left[a\right] = a$ $E[aX] = aE[X] = a\mu$ $E[aX + b] = aE[X] + E[b] = aE[x] + b = a\mu+b$

Knowing these rules, lets try out a challenge:

2.4.4 A challenge in Expected Values

Find $$E[\bar{x}]$$ where $$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}{x_i}$$

E.G find the expected value of an average

2.4.5 A solution

First, we can factor out the 1/n, so we have

$\frac{1}{n}E\left[\sum_{i=1}^{n}{x_i}\right]$

then we can say that the sum is just $$x_1+x_2+x_3+ ... + x_n$$. This would lead us to the conclusion that $E[\bar{x}] =\frac{1}{n}\sum_{i=1}^{n}E\left[x_i\right]$

If we let each $$x_i$$ be a constant then we have:

$E[\bar{x}] =\frac{1}{n}\sum_{i=1}^{n}x_i = \bar{x}$

This makes sense as $$\bar{x}$$ is the average value of x, and so the expected value of x for a very high n is the average value of x

2.5 Variance and Covariance

2.5.1 Variance

The variance is the dispersion. Variance is defined as $\mathrm{Var}\left(X\right) = E \left[\left(X-\mu\right)^2\right] = \int_{-\infty}^{\infty} \left(x - \mu\right)^{2}f\left(x\right)dx = \sigma^2$

$\widehat{\mathrm{Var}\left(X\right) }= \sum\left(x_i - \bar{x}\right)^2$

2.5.2 Covariance

Assume we have two RVs, $$X$$ and $$Y$$:

$\mathrm{Cov}(X,Y) = E\left[\left(X - \mu_x\right)\left(Y - \mu_y\right)\right]$

$\widehat{\mathrm{Cov}\left(X,Y\right) }= \sum\left(x_i - \bar{x}\right)\left(y_i - \bar{y}\right)$

The variance is, in essence, the sum of cross products.

2.5.3 Some generalizations on covariance:

If y tends to increase with x, then $$\mathrm{Cov}(X,Y)>0$$

If Y tends to decrease with X, then $$\mathrm{Cov}(X,Y)$$ is Less than 0

If it appears as a random cloud, then $$\mathrm{Cov}(X,Y)$$ is approximately 0

2.5.4 A challenge

Let $$X$$ be a RV. What is $$\mathrm{Cov}(X,X)$$?

2.5.5 A solution

$\mathrm{Cov}(X,X) = E\left[\left(X - \mu_x\right)\left(X - \mu_X\right)\right]$

We see that the arguments in the left and right parens are exactly the same. Thus, we can rewrite the equation as:

$\mathrm{Cov}(X,X) = E \left[\left(X - \mu_x\right)^2\right] = \mathrm{Var}(X)$

2.5.6 A note on covariance and correlation

Correlation is covariance but standardized.

$\mathrm{Corr}(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\mathrm{SD}(X)\mathrm{SD}(Y)}$

$=\frac{E[X - \mu_x)(Y-\mu_y)]}{\sigma_x \sigma_y}$

$\widehat{\mathrm{Corr}(X,Y)} = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{ns_x s_y}$