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Expected value is a fundamental concept in probability and statistics that represents the average outcome of a random variable if it were repeated many times. It is often denoted as \( E[X] \) for a random variable \( X \). The expected value can be thought of as the center of the distribution of the random variable.

The expected value takes into account not just the values a random variable can take, but also the probabilities of these values. This is why it is often called the 'weighted average'. The formula for calculating the expected value is:

• For discrete random variables: \( E[X] = \sum x_i P(x_i) \)
• For continuous random variables: \( E[X] = \int x f(x) dx \)

In our context, we use the linearity property of expectation to work with covariance, which helps to simplify expressions when dealing with linear transformations of random variables.

Variance measures how far a set of numbers (or a random variable) are spread out from their average value. It tells us about the distribution's dispersion, giving us insight into how much values in the dataset or random variable deviate from the mean. Mathematically, variance is represented as \( \operatorname{Var}(X) \) and calculated as:

• \( \operatorname{Var}(X) = E[(X - E[X])^2] \)
• This is equivalent to \( \operatorname{Var}(X) = E[X^2] - (E[X])^2 \)

When dealing with linear transformations, variance transforms in a specific way: \( \operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X) \). This means that linear scaling affects the dispersion based on the square of the scaling factor.

Comprehending variance helps in understanding the behavior of the data and its reliability. It also plays a crucial role in calculating standard deviation.

The standard deviation is a widely used measure of the amount of variation or dispersion of a set of values. It is the square root of variance, providing a metric that is in the same unit as the original data, making it more interpretable. It is denoted as \( \sigma \) for standard deviation. The standard deviation is calculated as:

• For a random variable \( X \), \( \sigma_X = \sqrt{\operatorname{Var}(X)} \)

The relationship between variance and standard deviation makes the latter a more intuitive measure for understanding data spread. Linear transformations also affect the standard deviation: when a random variable is scaled by a factor \( a \), its standard deviation becomes \( \sigma_{aX+b} = |a| \sigma_X \).

Understanding standard deviation is essential for interpreting correlation and is a key component in risk and relative volatility measurement.

Linear transformations involve scaling and translating random variables, typically expressed as \( aX + b \). In the context of covariance and correlation, these transformations alter the values of random variables but not certain relationships between them.For covariance, linear transformations adjust based on the product of the scaling factors. Hence, \( \operatorname{Cov}(aX+b, cY+d) = ac \operatorname{Cov}(X, Y) \). Importantly, the offset terms \( b \) and \( d \) do not affect the covariance because covariance is concerned solely with variability around the mean, not the actual values.Correlation, however, is a normalized form of covariance, independent of scale. For correlation, linear transformations do not change the correlation measure if the scaling factors \( a \) and \( c \) maintain the same sign. Therefore, the correlation remains consistent: \( \operatorname{Corr}(aX+b, cY+d) = \operatorname{Corr}(X, Y) \). Yet, if they have opposite signs, the correlation inverts: \( \operatorname{Corr}(aX+b, cY+d) = -\operatorname{Corr}(X, Y) \), reflecting the directional change in relationship.