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This is the repository of the formulas we use along other articles. The purpose of this document is to have in a summarized way, the mathematical operators and terms along with their notation and some useful relationships between them. We won't demonstrate how to obtain them. That is out of the scope of this document and there are many resources on the web about that.

We use the "⋅" operator to denote multiplication. This way, if we see the term "abxy", in a math expression, we know it represents a single entity and not —for example— the multiplication of the terms "ab" by "xy", in which case we will use "ab⋅xy".

In these formulas, the operator "==" is used mainly to show two alternate ways to express the same thing.

# Statistics Formulas

In these formulas, the symbols with bold typeface (e.g. **X**) represent random variables and the symbols with regular (non-bold) typeface, represent non-random variables (e.g. "*c*").

## The Expected Value

We use the expression *Eva( X )* to denote the

*Expected Value*of the random variable

**X**. The symbol μ

_{x}represents the value resulting from that expression.

The *Expected Value* of a non-random variable —for example a constant— is the value of that non-random variable itself:

The *Expected Value* of the product of two **independent** random variables is equal to the product of those variables *Expected value*s:

If the variable is scaled by a constant (a non-random variable) the *Expected Value* gets scaled by that constant. This is a direct consequence of the two previous formulas.

If two random variables are **correlated**, it means the value of one of them, in some degree, determines or influences the value of the other one. The *Covariance* is a measure of how much those variables are correlated.

For example, smoking is correlated with the probability of having cancer: the more you smoke, the greater the likelihood you eventually will get cancer.

The *Expected Value* of the product of two **correlated** random variables is equal to the product of those variables *Expected value*s plus the *Covariance* of them:

Notice how the formula 3 is a particular case of the previous formula: when the random variables are independent, the *Covariance* term is zero and goes away.

The *Expected Value* of the sum of **any** random variables is equal to the sum of the *Expected Value*s of those variables. This is true **regardless** if those random variables are independent or not.

The consolidation of the previous relationships results in the *Linearity* property of the *Expected Value* operator (*Linearity of Expectation*), which is true **regardless** the random variables are independent or not.

## The Covariance

We use the expression *Cov( X, Y )* to denote the

*Covariance*of the two random variables

**X**and

**Y**.

The *Covariance* is a measure of how much the values of each of two **correlated** random variables determines the other. If both variables change in the same way (e.g. when —in general— one grows the other also grows), the *Covariance* is positive, otherwise it is negative (e.g. when one increases the other decreases).

If the variables are **independent** the *Covariance* is zero. However, the *Covariance* must be considered zero or non zero from a statistical point of view, which means there is some fact leading us to believe the *Covariance* value can not be justified on the basis of chance alone.

If the variables are **independent**, their *Covariance* is zero. This can clearly be seen in the following formula, where if both variables were independent to each other, both terms in the subtraction become the same value and the *Covariance* becomes to zero:

However, the converse of the previous rule is not alway true: If the *Covariance* is zero, it **does not necessarily mean** the random variables are independent.

For example, if **X** is uniformly distributed in `[-1, 1]`

, its *Expected Value* and the *Expected Value* of the odd powers (e.g. **X**³) of **X** result zero in `[-1, 1]`

. For that reason, if the random variable **Y** is defined as **Y** = **X**², clearly **X** and **Y** are **correlated**. However, their *Covariance* is numerically equal to zero:

The *Covariance* is commutative.

The *Covariance* is invariant to the displacement of one or both variables.

The *Covariance* is scaled by each of the scales of X and Y.

Consolidating the two last equations:

## The Variance

The *Variance* is the special case of the *Covariance* when both variables are the same.

The *Variance* measures how much the values of a random variable are spread out. In other words, how much they are different among them.

We can compute the *Variance* from the *Expected Values* of **X** and its square (**X**²).

The *Variance* of a non-random variable is zero.

The *Variance* is invariant to the displacement of the variable.

If the variable is scaled by a constant (a non-random variable) the *Variance* gets scaled by the square of that constant.

The *Variance* of the sum or difference of two **correlated** random variables is:

If the variables are **independent**, the *Covariance* is zero and the *Variance* of the sum or the difference is the same:

The *Variance* of the product of two **correlated** random variables:

Using this relationship between the variance and the expected values we can say "*Eva(X²) = Var(X) + Eva²(X)*". Using that in the previous formula we get:

The *Variance* of the product of two **independent** random variables comes from the previous formulas, knowing that in such case $(\sigma_{X,Y} = \sigma_{X^2,Y^2} = 0)$:

Another way to get the product of two **independent** random variables is through this beautiful equation:

## The Standard Deviation

We use the expression *StdDev(X)* to denote the *Standard Deviation* of the random variable **X**.

The *Standard Deviation* is the square root of the *Variance*.

## The Correlation Coefficient

The Correlation Coefficient is a measure of dependence between two —correlated— random variables.

# Analytic Relationship between two Numerically Correlated Functions

Sometimes you can measure two output variables, from an experiment, which are strongly correlated because they come from two functions of a common variable and what you wanna know these two functions. If you build a correlation formula for the relationship between those output variables, with the help of data from many experiments, you can find a good approximation of that correlation formula. However, the question is: How this correlations formula can shed us light about the functions producing those output variables?

Let's suppose, we have $(f(x))$ and $(g(x))$, both evaluated at $(x_i)$ in $({x_1, x_2... x_n})$, which means we have the points $((f(x_i), g(x_i)))$. Now we find a correlation over those points. However, the correlation formula is, of course, between $(f())$ and $(g())$. How can we relate this correlation with $(f(x))$ and $(g(x))$?

To solve this question we will think we are plotting $((z, h(z)))$, then clearly:

\begin{align*} (z, h(z)) &= (f(x_i), g(x_i))\\ \\ z &= f(x) \\ h(z) &= g(x) \\ \\ x &= f^{-1}(z) \\ \therefore h(z) &= g(f^{-1}(z)) \\ \end{align*}

## Linear case

For example, let's see the case when both, f() and g(), functions are linear:

\begin{align*} ({\bf z}, h({\bf z})) &= (f({\bf x}), g({\bf x})) \\ \\ g({\bf x}) &= g_1 \cdot {\bf x} + g_0 \\ f({\bf x}) &= f_1 \cdot {\bf x} + f_0 \\ \\ {\bf x} &= f^{-1}({\bf z}) \\ {\bf x} &= \frac{{\bf z} - f_0}{f_1} \\ \\ h({\bf z}) &= g(f^{-1}({\bf z})) \\ h({\bf z}) &= g\left(\frac{{\bf z} - f_0}{f_1}\right) \\ h({\bf z}) &= g_1 \cdot \left(\frac{{\bf z} - f_0}{f_1}\right) + g_0 \\ \therefore h({\bf z}) &= \frac{g_1}{f_1} \cdot {\bf z} + \left( g_0 - \frac{g_1}{f_1} \cdot f_0 \right) \tag{32} \end{align*}

This means that when we correlate two linear functions, the resulting curve is also a line, having as slope the one of $(g(x))$ divided by the one of $(f(x))$. We also know the correlated line crosses the $(h)$ axis (when $(z = 0)$) at $(h = (g_1/f_1) \cdot z + ( g_0 - (g_1/f_1) \cdot f_0 ))$.

We can "validate" this case very easily. Let's suppose our linear functions are:

\begin{align*} f(x) &= 2 \cdot x + 13 \\ g(x) &= 3 \cdot x + 5 \end{align*}

If we plot these functions for $(x)$ in $({1, 2})$ we get $((f, g) = { (15, 8), (17, 11) })$. Knowing those points belong to a line, we can find the slope as $(\Delta y/ \Delta x)$. If our previous deduction is correct, this slope should be equal to $(g_1/f_1 = 3/2)$:

\begin{equation*} slope = \Delta y/\Delta x = (11 - 8) / (17 - 15) = 3/2 \end{equation*}

Which is exactly what we thought!

Also according to our last formula, the interception of the correlated line with $(z = 0)$ should occur when h is:

\begin{align*} h(0) &= (g_1/f_1) \cdot 0 + (g_0 - (g_1/f_1) \cdot f_0) \\ h &= 5 - (3/2) \cdot 13 \\ h &= -14.5 \end{align*}

We can find the interception with $(z = 0)$ from the correlated line data, knowing it contains the point (15, 8):

\begin{align*} (h - 8) &= (3/2) \cdot (z - 15) \\ (h - 8) &= (3/2) \cdot (0 - 15) \\ h &= 8 - (3/2) \cdot 15 \\ h &= -14.5 \end{align*}

Again the result is consistent to our deduction!

## Quadratic case

When $(f())$ is linear but $(g())$ is quadratic, we get:

\begin{alignat*}{2} f({\bf x}) &=& &f_1 \cdot {\bf x} + f_0& \cr g({\bf x}) &=&~~g_2 \cdot {\bf x}^2 ~+~ &g_1 \cdot {\bf x} + g_0& \cr \end{alignat*}

\begin{align*} {\bf z} &= f({\bf x}) \cr {\bf x} &= \frac{{\bf z} - f_0}{f_1} \cr \cr h({\bf z}) &= g_2 \cdot \left(\frac{{\bf z} - f_0}{f_1}\right)^2 + g_1 \cdot \left(\frac{{\bf z} - f_0}{f_1}\right) + g_0 \cr \end{align*}

Developing the last equation and summarizing, we finally get:

$$ ({\bf z}, h({\bf z})) = (f({\bf x}), g({\bf x})) $$

\begin{alignat*}{2} f({\bf x}) &=& &f_1 \cdot {\bf x} + f_0& \cr g({\bf x}) &=&~~g_2 \cdot {\bf x}^2 ~+~ &g_1 \cdot {\bf x} + g_0& \cr \end{alignat*}

$$ \begin{align*} \therefore h({\bf z}) &= z_2 \cdot {\bf z}^2 + z_1 \cdot {\bf z} + z_0 \cr \cr z_2 &= \frac{g_2}{f_1^2} \cr z_1 &= \frac{g_1}{f_1} - 2 \cdot \frac{f_0 \cdot g_2}{f_1^2} \cr z_0 &= \frac{g_2}{f_1^2} \cdot f_0^2 - \frac{g_1}{f_1} \cdot f_0 + g_0 \cr \end{align*}\tag{33} $$