# Covariance and Correlation

## Covariance

The covariance between two rv’s, X and Y, is defined as

\(\operatorname{Cov}(X, Y)=E[(X-E(X))(Y-E(Y))] = E[(X- \mu_x))(Y- \mu_y)]\)

**The covariance depends on both the set of possible pairs and the probabilities for those pairs.**

If both variables tend to deviate in the same direction (both go above their means or below their means at the same time), then the covariance will be positive.

If the opposite is true, the covariance will be negative.

If X and Y are not strongly (linearly) related, the covariance will be near 0.

### Computational formula for Covariance

\(\operatorname{Cov}(X, Y)=E[XY] -E[X]E[Y]\)

## Correlation Coefficient

The correlation Coefficient of X and Y , denoted by Cor(X, Y ) Represented by the Greek letter ‘’ρ’’ (rho)

\(Cor(X, Y) = \rho_{X,Y}= \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}\)

It represents a “scaled” covariance. The correlation is always between -1 and 1.

## Transformations of Distributions

### Discrete Distributions

Suppose that 𝖷 ∼ 𝖻𝗂𝗇(𝗇, 𝗉) What is the distribution of Y = n-X?

\(f(x)=P(X=x)= \binom{n}{x}p^x(1-p)^{n-x} \cdot I_{\{1,2,3, \ldots\}}(x)\)

**Just do it:**

#### Continuous Distributions

### Invertible functions

In the most general sense, are functions that “reverse” each other. For example, if f takes a to b, then the inverse, \(f^{-1}\) must take b to a. a function is invertible only if each input has a unique output. That is, each output is paired with exactly one input. That way, when the mapping is reversed, it will still be a function!

For X discrete or continuous, the cumulative distribution function (cdf) Is denoted by F(x) and is defined by

\(F(X)= P(X < x)\)