Random Variables: Difference between revisions
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= Linear Combinations of RV = | = Linear Combinations of RV = | ||
Let <math>X</math> be | Let <math>X,Y</math> be random variables, <math>a, b, c</math> be constants. | ||
Expectation have the following properties | Expectation have the following properties for linear transformations: | ||
<math> | <math> | ||
E(aX + c) = aE(X) + c | E(aX + c) = aE(X) + c | ||
</math> | |||
For a linear combination, we have | |||
<math> | |||
E(aX + bY) = aE(X) + bE(Y) | |||
</math> | </math> | ||
Variance is a bit more complicated. Recall that the calculation of | Variance is a bit more complicated. Recall that the calculation of | ||
variance involves the average difference from the mean squared. | 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. | |||
<math> | <math> | ||
Var(aX + c) = a^2 Var(X) | Var(aX + c) = a^2 Var(X) | ||
</math> | |||
For a linear combination, when the two events are independent, | |||
<math> | |||
Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) | |||
</math> | |||
When the events are dependent, | |||
<math> | |||
Var(aX + bY) = a^2 Var(X) + ab Cov(X,Y) + b^2 Var(Y) | |||
</math> | </math> | ||
Revision as of 01:45, 2 March 2024
Linear Combinations of RV
Let be random variables, be constants.
Expectation have the following properties for linear transformations:
For a linear combination, we have
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.
For a linear combination, when the two events are independent,
When the events are dependent,
