Linear Combinations of Gaussian Random Variables

Summary

A linear combination forms a new random variable by multiplying each input variable by a constant and adding the results. The document gives examples with different coefficients and an added constant, then expresses the general form as a weighted sum. Multiplying variables by each other, squaring them, or taking roots is outside this definition.

The explanation calls the inputs Gaussian, or normally distributed, but does not discuss how their dependence affects the distribution of the sum. A weighted sum is guaranteed to be Gaussian when the variables are jointly Gaussian; Gaussian marginal distributions alone do not guarantee that result. The note is therefore useful for understanding the basic operation, but it does not cover the conditions for the combined variable to remain Gaussian, or implications for modeling or trading.

Key ideas

• A linear combination multiplies random variables by constants and adds the resulting terms.
• The coefficients may be positive, negative, or zero.
• Combining variables this way does not include multiplying the variables by one another or applying powers or roots.
• A weighted sum is Gaussian when its inputs are jointly Gaussian, a condition the document does not explain.

Tags

# Linear combination of gaussian random variables


# Linear combination of gaussian random variables












I know what random variables are but I don't understand what a linear combination of gaussian random variables is. Can anyone please give me an explanation or clues? Thanks in advance, Julien.

## Answer by Lepto Kurtič (score 3, accepted)

https://quant.stackexchange.com/a/1253

Gaussian random variable is another name for Normal random variable. It is called Gaussian because Carl Friedrich Gauss discovered many properties of the Normal distribution.

A linear combination of Gaussian random variables is another random variable, not necessarily Gaussian itself, that you get by adding and subtracting Gaussian random variables. Lets call this linear combination of Gaussian variables $Y$. The random variable $Y$ could be something like $$ Y = 2 \times X_1 + 1.2234 \times X_2 -7 $$ Or it could be something like $$Y = \frac{X_1 - 0.01 \times X_2}{12}$$

where $X$'s are Gaussian random variables. As you can see a linear combination of them is obtained by summing them up or subtracting them from each other, but never multiplying or dividing them with each other or by itself (like squaring or taking roots).

We can give general form of linear combination of random Gaussian variables: $$ Y = a_1X_1 + a_2X_2 + ... + a_n X_n$$

where $a_i$ is any number you want it to be, like -222, or 0 or 17.222...

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.