2.2.2. Moments

**Proposition 2.** *The rth moments of X* ∼ MPN (0, 1, *α*) *for r* = 1, 2, 3, . . . , *are given by*

$$\mathbb{E}(X^r) = \frac{\alpha}{2^{\alpha} - 1} a\_r(\alpha), \tag{11}$$

.

*where ar*(*α*) *is defined as*

$$a\_I(u) = \int\_0^1 [\Phi^{-1}(u)]^r (1+u)^{a-1} \, du. \tag{12}$$

*Here,* <sup>Φ</sup>−<sup>1</sup>(·) *is the quantile function of the standard normal distribution.*

**Proof.** By using the change of variable *u* = <sup>Φ</sup>(*x*), it follows that

$$\mathbb{E}(X^r) \quad = \int\_{-\infty}^{\infty} x^r \frac{\alpha}{2^{\alpha} - 1} \phi(x) [1 + \Phi(x)]^{\alpha - 1} \, dx$$

$$= \frac{\alpha}{2^{\alpha} - 1} \int\_{0}^{1} [\Phi^{-1}(u)]^r (1 + u)^{\alpha - 1} \, du$$

$$= \frac{\alpha}{2^{\alpha} - 1} a\_r(\alpha) . \square$$

**Corollary 1.** *The mean and variance of X are given by*

$$\begin{array}{rcl} \mathbb{E}(X) &=& \frac{\alpha}{2^{\alpha}-1}a\_1(\alpha) \quad \text{and} \\ \mathbb{V}ar(X) &=& \frac{\alpha}{2^{\alpha}-1} \left(a\_2(\alpha) - \frac{\alpha}{2^{\alpha}-1}a\_1^2(\alpha)\right), \end{array}$$

*respectively.*

**Corollary 2.** *The skewness (β*1*) and kurtosis (β*2*) coefficients are, respectively, given by*

$$\begin{array}{rcl} \beta\_{1} &=& \frac{a\_{3}(a) - \frac{3a}{2^{t}-1}a\_{1}(a)a\_{2}(a) + \frac{2a^{2}}{(2^{t}-1)^{2}}a\_{1}^{3}(a)}{\left(\frac{a}{2^{t}-1}\right)^{3/2}\left[a\_{2}(a) - \frac{a}{2^{t}-1}a\_{1}^{2}(a)\right]^{3/2}} & \text{ and} \\\\ \beta\_{2} &=& \frac{a\_{4}(a) - \frac{4a}{2^{t}-1}a\_{1}(a)a\_{3}(a) + \frac{6a^{2}}{(2^{t}-1)^{2}}a\_{1}^{2}(a)a\_{2}(a) - \frac{3a^{3}}{(2^{t}-1)^{3}}a\_{1}^{4}(a)}{\frac{a}{2^{t}-1}\left[a\_{2}(a) - \frac{a}{2^{t}-1}a\_{1}^{2}(a)\right]^{2}}. \end{array}$$

**Remark 4.** *Observe that the integral in Equation (12) can be numerically approximated by using the built-in function integrate available in the software package* R*. Below, in Table 2, some approximations of the mean and* *variance for the* MPN *distribution for different values of α are displayed. Figure 3 illustrates the behavior of the* E(*X*) *and* V*ar*(*X*) *of the* MPN *distribution for different values of α. It is observable that when α grows, the mean increases and the variance decreases.*

*Figure 4 displays the curves associated with the coefficients of skewness (left panel) and kurtosis (right) of the* MPN *and* PN *distributions. It is shown that, depending on the values of α, the* MPN *distribution exhibits equal, greater, or lesser values for these coefficients compared to the* PN *model. In general, the* MPN *distribution has a smaller range of skewness than the* PN *distribution. On the other hand, when α* < 13.05*, the* MPN *distribution has a greater kurtosis coefficient than the* PN *model.*

**Table 2.** Approximations of E(*X*) and V*ar*(*X*) of the MPN distribution for different values of *α*.


**Figure 3.** Plot of the E(*X*) and V*ar*(*X*) of the MPN distribution.

**Figure 4.** Graphs of the skewness and kurtosis coefficients for the MPN and PN distributions.
