# 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.

-

## 1 Answer

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...

-

1
Thanks for this detailed answer Hassmann, the reason I asked this question is that according to my book, when a vector of risk factors is gaussian one can use Cholesky factorization; when it is not one has to use copulas for one's monte carlo simulations. Say I have three factors: underlying stock, stochastic IR and volatility. How do I know whether my vector is gaussian or not? – balteo May 31 '11 at 11:55
You can look at the wikipedia page on multivariate normal distribution. There is a paragraph on multivariate normality test. – Zarbouzou May 31 '11 at 17:03
Thanks then to both of you!! – Julien May 31 '11 at 18:24
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9020764231681824, "perplexity_flag": "head"}