# Monomial distribution of $X^a \cdot Y^b$

What is the distribution of the following monomial? $$X^a \cdot Y^b$$ where $X$ and $Y$ are normal random variables and $a$ and $b$ are natural numbers.

For example, when $X \sim N(0,1)$, $a=2$, and $b=0$ it is a Chi-squared distribution, which has a variance of 2.

What if we have $n$ independent variables $X_1, X_2, \dots , X_n$, with $X_i \sim N(0,\sigma^2)$ and some natural numbers $p_1, p_2, \dots,p_n$. What can we say about the variance of the following r.v.?

$$X_1^{p_1} \cdot X_2^{p_2} \cdots X_n^{p_n}$$

-

1
– user10525 May 11 '12 at 8:28

## 1 Answer

The first question about a distribution has no convenient general answer, because AFAIK nobody has assigned names to such distributions nor extensively studied and characterized them except when both $a$ and $b$ are $2$ or less.

Concerning the second question about the variances, as shorthand write $\mathbf{p}=(p_1,p_2,\ldots,p_n)$ and $\mathbf{x^p} = X_1^{p_1} X_2^{p_2} \cdots X_n^{p_n}$.

Recall that for the standard Normal distribution the $k^\text{th}$ moment is $0$ when $k$ is odd and otherwise equals $(k-1)!! = (k-1)(k-3)\cdots(3)(1)$. From this and the independence assumption, the expression for the expectation of $\mathbf{x^p}$ is immediate: it equals $0$ when one or more of the $p_i$ is odd and otherwise is the product of the $(p_i-1)!!$, which I will similarly abbreviate $(\mathbf{p-1})!!$.

By definition,

$$\text{Var}(\mathbf{x^p}) = \mathbb{E}[(\mathbf{x^p})^2] - \mathbb{E}[\mathbf{x^p}]^2 = (2\mathbf{p}\mathbf{-1})!! - ((\mathbf{p-1})!!)^2.$$

When the variables are scaled to have variance $\sigma^2$, $\mathbf{x^p}$ will be multiplied by $|\sigma|^{p_1+p_2+\cdots+p_n}$, whence its variance will be multiplied by $\sigma^{2(p_1+p_2+\cdots+p_n)}$.

### Example

Let $\mathbf{p} = (2,4)$:

• $(2\mathbf{p}\mathbf{-1})!! = 3!! 7!! = [3(1)][7(5)(3)(1)] = 315$;

• $((\mathbf{p-1})!!)^2 = 1!!3!! = ([1][3(1)])^2 = 9$;

• $\text{Var}(X_1^2 X_2^4) = (315 - 9)\sigma^{2(2+4)} = 306\sigma^{12}.$

-
{"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": 34, "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": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9223197102546692, "perplexity_flag": "head"}