Agenda

<aside> 📌 Reminders

• Quiz solution presentation </aside>

Covariance

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Covariance: measures the joint variability of two random variables $X$ and $Y$,

$$ \text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]  $$

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Key points

• Positive covariance indicates that when $X$ increases, $Y$ tends to increase as well.
• Negative covariance indicates that when $X$ increases, $Y$ tends to decrease.
• A zero covariance implies no linear relationship between $X$ and $Y$.

Correlation

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Correlation: a normalized form of covariance that measures the strength and direction of a linear relationship between two variables $X$ and $Y$, it is given by

$$ \rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}  $$

where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$ respectively

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Key Points:

• The correlation coefficient $\rho$ ranges from $-1$ to $1$:
  • $\rho = 1$: Perfect positive correlation.
  • $\rho = -1$: Perfect negative correlation.
  • $\rho = 0$: No linear correlation.

Cauchy-Schwarz Inequality

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Cauchy-Schwarz inequality: states that for any two real-valued random variables $X$ and $Y$, the following inequality holds:

$$ \left( \mathbb{E}[XY] \right)^2 \leq \mathbb{E}[X^2] \mathbb{E}[Y^2]  $$

i.e. the covariance between two variables is bounder by their individual variances

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Markov Inequality

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Markov inequality: a probabilistic bound that applies to any non-negative random variable $X$. It states that $\forall a > 0$:

$$ P(X \geq a) \leq \frac{\mathbb{E}[X]}{a}  $$

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• Very useful for obtaining rough bounds when little is known about the distribution of $X$, other than its expectation.

Practice