Random Variables

Linear Combinations of RV

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X,Y} be random variables, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c} be constants.

Expectation have the following properties for linear transformations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(aX + c) = aE(X) + c }

For a linear combination, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(aX + bY) = aE(X) + bE(Y) }

Variance is a bit more complicated. Recall that the calculation of variance involves the average difference from the mean squared. It is no surprise that any constant coefficient is squared, and any translations does not impact the spread of data.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Var(aX + c) = a^2 Var(X) }

For a linear combination, when the two events are independent,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) }

When the events are dependent,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Var(aX + bY) = a^2 Var(X) + ab Cov(X,Y) + b^2 Var(Y) }