# Approximating the Distribution of the Product of Two Normally Distributed Random Variables

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## Abstract

**:**

## 1. Introduction

## 2. Product of Two Normal Random Variables

- Mean: $\mathrm{E}\left(U\right)={\delta}_{X}{\delta}_{Y}+\rho $.
- Variance: $\mathrm{Var}\left(U\right)=1+{\rho}^{2}+2\rho {\delta}_{X}{\delta}_{Y}+{\delta}_{X}^{2}+{\delta}_{Y}^{2}$.
- Coefficient of variation: $\mathrm{CV}\left(U\right)={\displaystyle \frac{\sqrt{1+{\rho}^{2}+2\rho {\delta}_{X}{\delta}_{Y}+{\delta}_{X}^{2}+{\delta}_{Y}^{2}}}{{\delta}_{X}{\delta}_{Y}+\rho}}$.
- Coefficient of skewness: $\mathrm{CS}\left(U\right)={\displaystyle \frac{2({\rho}^{3}+3{\delta}_{X}{\delta}_{Y}+3{\rho}^{2}{\delta}_{X}{\delta}_{Y}+3\rho (1+{\delta}_{X}^{2}+{\delta}_{Y}^{2}))}{{(1+{\rho}^{2}+{\delta}_{X}^{2}+{\delta}_{Y}^{2}+2\rho {\delta}_{X}{\delta}_{Y})}^{3/2}}}$.
- Excess of kurtosis: $\mathrm{EK}\left(U\right)={\displaystyle \frac{6(1+{\rho}^{4}+2{\delta}_{X}^{2}+12\rho {\delta}_{X}{\delta}_{Y}+4{\rho}^{3}{\delta}_{X}{\delta}_{Y}+2{\delta}_{Y}^{2}+6{\rho}^{2}(1+{\delta}_{X}^{2}+{\delta}_{Y}^{2}))}{{(1+{\rho}^{2}+{\delta}_{X}^{2}+{\delta}_{Y}^{2}+2\rho {\delta}_{X}{\delta}_{Y})}^{2}}}.$
- Skewness/kurtosis ratio:$$\mathrm{SKR}\left(U\right)=\frac{\sqrt{1+{\delta}_{X}^{2}+{\delta}_{Y}^{2}+2{\delta}_{X}{\delta}_{Y}\rho +{\rho}^{2}}\left(\right)open="("\; close=")">3{\delta}_{X}{\delta}_{Y}+3(1+{\delta}_{X}^{2}+{\delta}_{Y}^{2})\rho +3{\delta}_{X}{\delta}_{Y}{\rho}^{2}+{\rho}^{3}}{}3\left(\right)open="("\; close=")">1+2{\delta}_{Y}^{2}+6(1+{\delta}_{Y}^{2}){\rho}^{2}+{\rho}^{4}+4{\delta}_{X}{\delta}_{Y}\rho (3+{\rho}^{2})+{\delta}_{X}^{2}(2+6{\rho}^{2}).$$

- (i)
- First factor $\sqrt{1+{\delta}_{X}^{2}+{\delta}_{Y}^{2}+2{\delta}_{X}{\delta}_{Y}\rho +{\rho}^{2}}$: its minimum value is one, which is reached at $\rho ={\delta}_{X}={\delta}_{Y}=0$, while its maximum value is $\sqrt{6}$ and reached at $\rho =1,{\delta}_{X}={\delta}_{Y}=-1$. Afterwards, the first factor ranges between $[1,\sqrt{6}]$.
- (ii)
- Second factor $\left(\right)$: its minimum value is $-16$ and reached when $\rho =-1,{\delta}_{X}=-1,{\delta}_{Y}=1$, while its maximum is 16 at $\rho =1,{\delta}_{X}=-1,{\delta}_{Y}=-1$. Thus, the second factor ranges between $[-16,16]$.
- (iii)
- Third factor $3(1+2{\delta}_{Y}^{2}+6(1+{\delta}_{Y}^{2}){\rho}^{2}+{\rho}^{4}+4{\delta}_{X}{\delta}_{Y}\rho (3+{\rho}^{2})+{\delta}_{X}^{2}(2+6{\rho}^{2}))$: its minimum value is three and reached at $\rho ={\delta}_{X}={\delta}_{Y}=0$, while its maximum value is 120 at $\rho =1,{\delta}_{X}={\delta}_{Y}=-1$. Hence, the denominator of this skewness/kurtosis ratio is in the range $[3,120]$.

## 3. The Extended Skew-Normal Distribution

- Mean: $\mathrm{E}\left(X\right)=\xi +\omega \sqrt{{\displaystyle \frac{2{\alpha}^{2}}{\pi (1+{\alpha}^{2})}}}$;
- Variance: $\mathrm{Var}\left(X\right)=\omega \left(\right)open="("\; close=")">1-{\displaystyle \frac{2{\alpha}^{2}}{\pi (1+{\alpha}^{2})}}$.

- Mean: $\mathrm{E}\left(Y\right)=\xi +{\displaystyle \frac{exp(-{\tau}^{2}/2)\alpha \omega \sqrt{2/\pi}}{2\mathsf{\Phi}\left(\tau \right)\sqrt{1+{\alpha}^{2}}}}.$
- Variance: $\mathrm{Var}\left(Y\right)={\omega}^{2}\left(\right)open="("\; close=")">1-{\displaystyle \frac{exp(-{\tau}^{2}){\alpha}^{2}\left(\right)open="("\; close=")">2+exp\left({\tau}^{2}/2\right)2\tau \mathsf{\Phi}\left(\tau \right)\sqrt{2\pi}}{}\pi (1+{\alpha}^{2}){\left(2\mathsf{\Phi}\left(\tau \right)\right)}^{2}}.$
- Coefficient of variation: $\mathrm{CV}\left(Y\right)={\displaystyle \frac{\sqrt{{\omega}^{2}\left(\right)open="("\; close=")">1-\frac{exp(-{\tau}^{2}){\alpha}^{2}\left(\right)open="("\; close=")">2+exp\left({\tau}^{2}/2\right)2\tau \mathsf{\Phi}\left(\tau \right)\sqrt{2\pi}}{}\pi \left(\right)open="("\; close=")">1+{\alpha}^{2}{\left(2\mathsf{\Phi}\left(\tau \right)\right)}^{2}}}{}}$
- Coefficient of skewness:$$\mathrm{CS}\left(Y\right)=\frac{exp\left(-\frac{3{\tau}^{2}}{2}\right){\alpha}^{3}\left(\right)open="("\; close=")">4\sqrt{\frac{2}{\pi}}+6exp\left(\frac{{\tau}^{2}}{2}\right)2\tau \mathsf{\Phi}\left(\tau \right)+exp\left({t}^{2}\right)\sqrt{2\pi}\left(\right)open="("\; close=")">{\tau}^{2}-1}{{\left(2\mathsf{\Phi}\left(\tau \right)\right)}^{2}}3/2$$
- Excess of kurtosis:$$\mathrm{EK}\left(Y\right)=\frac{exp\left(-2{\tau}^{2}\right){\alpha}^{4}\left(\right)open="("\; close=")">-24-24exp\left(\frac{{\tau}^{2}}{2}\right)\sqrt{2\pi}2\tau \mathsf{\Phi}\left(\tau \right)-2exp\left({t}^{2}\right)\pi \left(\right)open="("\; close=")">7{\tau}^{2}-4}{{\left(2\mathsf{\Phi}\left(\tau \right)\right)}^{2}}{(2\mathsf{\Phi}\left(\tau \right)-2)}^{3}{\pi}^{2}{(1+{\alpha}^{2})}^{2}{\left(2\mathsf{\Phi}\left(\tau \right)\right)}^{4}{\left(\right)}^{-}$$
- Skewness/kurtosis ratio:$$\begin{array}{ccc}\mathrm{SKR}\left(Y\right)\hfill & =& \frac{exp({\tau}^{2}/2)\pi \sqrt{1+{\alpha}^{2}}\mathsf{\Phi}{\left(\tau \right)}^{2}\left(\right)open="("\; close=")">\sqrt{2/\pi}+3exp({\tau}^{2}/2)\tau +exp\left({\tau}^{2}\right)\sqrt{2\pi}({\tau}^{2}-1)\mathsf{\Phi}{\left(\tau \right)}^{2}}{}\left(\right)open="("\; close=")">-3+\sqrt{2}exp\left(\frac{3{\tau}^{2}}{2}\right){\pi}^{3/2}\tau ({\tau}^{2}-3){(\mathsf{\Phi}(-\tau )-1)}^{3}-6exp({\tau}^{2}/2)\sqrt{2\pi}\tau \mathsf{\Phi}\left(\tau \right)-exp\left({\tau}^{2}\right)\pi (7{\tau}^{2}-4)\mathsf{\Phi}{\left(\tau \right)}^{2}\hfill \end{array}& & \times \frac{1}{\alpha}\sqrt{1-\frac{exp(-{\tau}^{2}){\alpha}^{2}\left(\right)open="("\; close=")">1+exp({\tau}^{2}/2)\sqrt{2\pi}\tau \mathsf{\Phi}\left(\tau \right)}{}2\pi ({\alpha}^{2}+1)\mathsf{\Phi}{\left(\tau \right)}^{2}}.\hfill $$

## 4. Approximation for the Product Distribution with ESN Distributions

- As non-linear equations are generated, suitable numerical methods are required.
- As the solution obtained can be sensitive to the initial point, different points can be tested to avoid the stop at an inadequate point.
- Values of the coefficient of skewness and the excess of kurtosis are not always reached.

- A known distribution is employed and its PDF for any value of the product can be calculated with no need of numerical integrals.
- Inference using the ESN distribution is available.

`R`software was used [30]. Specifically, the

`sn`package [31] was utilized, which provides functions for the SN and ESN PDFs and their cumulants. These cumulants are employed to derive the values of skewness and kurtosis.

`BB`package [32] was used, which contains the

`BBSolve`function, in order to solve the system of equations described above.

## 5. Conclusions and Future Research

- (i)
- An approximation was proposed for the distribution of the product of two normally distributed random variables while using the extended skew-normal distribution.
- (ii)
- The moment generating function for the product of two normally distributed variables was considered as a function of the inverse coefficients of variation of both variables and of the corresponding correlation. From this moment generating function, the mean, variance, skewness, and kurtosis of this product were calculated.
- (iii)
- The probability density function of an extended skew-normal distribution is known and, using its cumulants, expressions for the mean, variance, skewness, and kurtosis were calculated. Skewness and kurtosis depended only on two parameters, while the mean and variance are functions of the four parameters of the extended skew-normal distribution.
- (iv)
- A numerical evaluation of the proposed approach was considered, which showed its potential for applications.
- (v)
- A graphical comparison of the proposed approach with other two approaches proposed in the recent literature on the topic was carried out showing the superiority of our approach.

- (i)
- The approximation to the distribution of the product of two normally distributed variables, while using the extended skew-normal distribution, showed some problems when the inverse coefficients of variation both are in the range $[-1,1]$. For this range, the adequate values of skewness and kurtosis cannot be reached and, thus, the approximation does not work well there. Some further research should be conducted in this line in order to improve our approach.
- (ii)
- (iii)
- (iv)
- If there are two random samples, the first one from the distribution of the product of two normally distributed random variables, and the second one from the fitted extended skew-normal distribution, and then some non-parametric tests can be used to compare them. Moreover, a sophisticated approach, as, for example, bootstrapping, can be applied in this context.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Plots of the skewness/kurtosis ratio of the product of two normal variables for indicated values of $\rho $ and with inverse coefficients of variation ${\delta}_{X}\in [-5,5]$ and ${\delta}_{Y}\in [-5,5]$.

**Figure 2.**Plots of the skewness/kurtosis ratio of the product of two normal variables for indicated values of $\rho $ and with inverse coefficients of variation ${\delta}_{X}\in [-1,1]$ and ${\delta}_{Y}\in [-1,1]$.

**Figure 3.**Plots of the ESN skewness/kurtosis ratio with $\alpha \in [-10,10]$ and $\tau \in [-5,5]$.

**Figure 4.**Plots of the ESN skewness/kurtosis ratio in function of $\tau $ when (

**a**) $\alpha >0$ and (

**b**) $\alpha <0$.

**Figure 5.**PDF of the product of two normally distributed variables with: (red) simulation approach from $X\sim \mathrm{N}(1.0,2.0)$ and $Y\sim \mathrm{N}(0.5,2.0)$, for $\rho =0.5$, ${\delta}_{X}=0.5$ and ${\delta}_{Y}=0.25$; (blue) ESN approach; and (green) Pearson type III approach.

**Figure 6.**PDF of the product of two normally distributed variables with: (red) simulation approach from $X\sim \mathrm{N}(1.0,0.5)$ and $Y\sim \mathrm{N}(0.5,0.2)$, for $\rho =0.5$, ${\delta}_{X}=2.0$ and ${\delta}_{Y}=2.5$; (blue) ESN approach; (green) Pearson type III approach; and (black) Cui et al. approach.

**Figure 7.**PDF of the product of two normally distributed variables with: (red) simulation approach from $X\sim \mathrm{N}(1.0,2.0)$ and $Y\sim \mathrm{N}(0.5,0.2)$, for $\rho =0.5$, ${\delta}_{X}=0.5$ and ${\delta}_{Y}=2.5$; (blue) ESN approach; (green) Pearson type III approach; and (black) Cui et al. approach.

**Table 1.**Statistics for the product of two normally distributed variables: X and Y with $\rho =0.5$ and ${\delta}_{X},{\delta}_{Y}$ as listed.

Product | Mean | Variance | Skewness | Kurtosis |
---|---|---|---|---|

${\delta}_{X}=0.5,{\delta}_{Y}=0.25$ | ||||

Theoretical value | 2.500 | 27.0000 | 2.3338 | 9.25995 |

Simulated value | 2.499 | 26.9927 | 2.3337 | 9.40562 |

Approximate value | 2.500 | 27.0000 | 1.9855 | 9.26592 |

${\delta}_{X}=2.0,{\delta}_{Y}=2.5$ | ||||

Theoretical value | 0.5500 | 1.650 | 1.06679 | 1.5634 |

Simulated value | 0.5499 | 1.649 | 1.06684 | 1.5645 |

Approximate value | 0.5500 | 1.650 | 1.06677 | 1.5634 |

${\delta}_{X}=0.5,{\delta}_{Y}=2.5$ | ||||

Theoretical value | 0.7000 | 1.4400 | 1.18981 | 2.479685 |

Simulated value | 0.6999 | 1.4396 | 1.18956 | 2.479581 |

Approximate value | 0.7000 | 1.4400 | 1.18981 | 2.479685 |

**Table 2.**Comparison of statistics of the product of two normality distributed variables $X\sim \mathrm{N}({\mu}_{X},{\sigma}_{X})$ and $Y\sim \mathrm{N}({\mu}_{Y},{\sigma}_{Y})$ with indicated $\rho $ and ESN approximation statistics.

Theoretical Value | Approximated Value | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathit{X}}$ | ${\mathit{\delta}}_{\mathit{Y}}$ | Mean | Variance | Skewness | Kurtosis | Mean | Variance | Skewness | Kurtosis | |

$\rho =0.5$ | ||||||||||

$X\sim \mathrm{N}(2,1.75)$ $Y\sim \mathrm{N}(1,0.25)$ | 1.1 | 4.0 | 2.2188 | 4.4268 | 0.8043 | 1.0415 | 2.2188 | 4.4268 | 0.8043 | 1.0415 |

$X\sim \mathrm{N}(2,1.50)$ $Y\sim \mathrm{N}(1,0.50)$ | 1.3 | 2.0 | 2.3750 | 5.4531 | 1.3445 | 2.5612 | 2.3750 | 5.4531 | 1.3445 | 2.5612 |

$X\sim \mathrm{N}(2,1.25)$ $Y\sim \mathrm{N}(1,0.75)$ | 1.6 | 1.3 | 2.4688 | 6.7861 | 1.5038 | 3.1815 | 2.4688 | 6.7861 | 1.5038 | 3.1815 |

$X\sim \mathrm{N}(2,1.00)$ $Y\sim \mathrm{N}(1,1.00)$ | 2.0 | 1.0 | 2.5000 | 8.2500 | 1.4032 | 2.9146 | 2.5000 | 8.2500 | 1.4032 | 2.9146 |

$X\sim \mathrm{N}(2,0.75)$ $Y\sim \mathrm{N}(1,1.25)$ | 2.7 | 0.8 | 2.4688 | 9.7861 | 1.1440 | 2.1081 | 2.4688 | 9.7861 | 1.1440 | 2.1081 |

$X\sim \mathrm{N}(2,0.50)$ $Y\sim \mathrm{N}(1,1.50)$ | 4.0 | 0.7 | 2.3750 | 11.4531 | 0.7900 | 1.1209 | 2.3750 | 11.4531 | 0.7900 | 1.1209 |

$X\sim \mathrm{N}(2,0.25)$ $Y\sim \mathrm{N}(1,1.75)$ | 8.0 | 0.6 | 2.2188 | 13.4268 | 0.3924 | 0.3139 | 2.2188 | 13.4268 | 0.3923 | 0.3139 |

$\rho =-0.5$ | ||||||||||

$X\sim \mathrm{N}(1,0.25)$ $Y\sim \mathrm{N}(2,1.75)$ | 4.0 | 1.1 | 1.7813 | 2.6768 | −0.3993 | 1.0253 | 1.7813 | 2.6768 | −0.4075 | 1.0254 |

$X\sim \mathrm{N}(1,0.50)$ $Y\sim \mathrm{N}(2,1.50)$ | 2.0 | 1.3 | 1.6250 | 2.4531 | −0.0641 | 1.7198 | 1.6250 | 2.4531 | −0.0640 | 1.7196 |

$X\sim \mathrm{N}(1,0.75)$ $Y\sim \mathrm{N}(2,1.25)$ | 1.3 | 1.6 | 1.5313 | 3.0361 | −0.0410 | 1.9499 | 1.5313 | 3.0361 | −0.0412 | 1.9527 |

$X\sim \mathrm{N}(1,1.00)$ $Y\sim \mathrm{N}(2,1.00)$ | 1.0 | 2.0 | 1.5000 | 4.2500 | −0.3709 | 2.3460 | 1.5000 | 4.2500 | −0.3709 | 2.3460 |

$X\sim \mathrm{N}(1,1.25)$ $Y\sim \mathrm{N}(2,0.75)$ | 0.8 | 2.7 | 1.5313 | 6.0361 | −0.5836 | 2.0131 | 1.5313 | 6.0361 | −0.5853 | 2.0139 |

$X\sim \mathrm{N}(1,1.50)$ $Y\sim \mathrm{N}(2,0.5)$ | 0.7 | 4.0 | 1.6250 | 8.4531 | −0.5593 | 1.1367 | 1.6250 | 8.4531 | −0.5593 | 1.1367 |

$X\sim \mathrm{N}(1,1.75)$ $Y\sim \mathrm{N}(2,0.25)$ | 0.6 | 8.0 | 1.7813 | 11.6768 | −0.3399 | 0.3192 | 1.7813 | 11.6770 | −0.3399 | 0.3290 |

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**MDPI and ACS Style**

Seijas-Macías, A.; Oliveira, A.; Oliveira, T.A.; Leiva, V.
Approximating the Distribution of the Product of Two Normally Distributed Random Variables. *Symmetry* **2020**, *12*, 1201.
https://doi.org/10.3390/sym12081201

**AMA Style**

Seijas-Macías A, Oliveira A, Oliveira TA, Leiva V.
Approximating the Distribution of the Product of Two Normally Distributed Random Variables. *Symmetry*. 2020; 12(8):1201.
https://doi.org/10.3390/sym12081201

**Chicago/Turabian Style**

Seijas-Macías, Antonio, Amílcar Oliveira, Teresa A. Oliveira, and Víctor Leiva.
2020. "Approximating the Distribution of the Product of Two Normally Distributed Random Variables" *Symmetry* 12, no. 8: 1201.
https://doi.org/10.3390/sym12081201