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This question is related to the $g_i(x)$ functions which are defined below in terms of the $f_i(x)$ functions defined in my previous question at ref(1) where all $f_i(x)$ functions are evaluated with the coefficient function $a(n)=1$. The definitions below are for $x>0$, and assume the definition $g_i(x)=0$ for $xle 0$.

(1) $quad g_1(x)=frac{2}{sqrt{pi}},f_1'(x),,quad x>0$

(2) $quad g_1'(x)=frac{2}{sqrt{pi}},f_1''(x),,quad x>0$

(3) $quad g_2(x)=2,f_2'(x),,qquad x>0$

(4) $quad g_2'(x)=2,f_2''(x),,qquad x>0$

(5) $quad g_5(x)=2,f_5(x),,qquad x>0$

(6) $quad g_5'(x)=2,f_5'(x),,qquad x>0$

(7) $quad g_6(x)=2,f_6(x),,qquad x>0$

(8) $quad g_6'(x)=2,f_6'(x),,qquad x>0$

Expanded definitions of the $g_i(x)$ functions are provided below. Again, the definitions below are for $x>0$, and assume the definition $g_i(x)=0$ for $xle 0$.

(9) $quad g_1(x)=frac{4}{sqrt{pi},x^3}sum_limits{n=1}^infty n^2,e^{-frac{n^2}{x^2}},,quad x>0$

(10) $quad g_1'(x)=frac{4}{sqrt{pi},x^6}sumlimits_{n=1}^infty n^2 left(2,n^2-3,x^2right) e^{-frac{n^2}{x^2}},,quad x>0$

(11) $quad g_2(x)=frac{4,pi}{x^3}sumlimits_{n=1}^infty n^2,e^{-frac{pi,n^2}{x^2}},,quad x>0$

(12) $quad g_2'(x)=frac{4,pi}{x^6}sumlimits_{n=1}^infty n^2left(2,pi,n^2-3,x^2right) e^{-frac{pi,n^2}{x^2}},,quad x>0$

(13) $quad g_5(x)=2sumlimits_{n=1}^inftyleft(frac{2,n^2}{x^2}-1right),e^{-frac{n^2}{x^2}},,quad x>0$

(14) $quad g_5'(x)=frac{4}{x^5}sumlimits_{n=1}^infty n^2,left(2,n^2-3,x^2right),e^{-frac{n^2}{x^2}},,quad x>0$

(15) $quad g_6(x)=2sumlimits_{n=1}^inftyleft(frac{2,pi,n^2}{x^2}-1right),e^{-frac{pi,n^2}{x^2}},,quad x>0$

(16) $quad g_6'(x)=frac{4,pi}{x^5}sumlimits_{n=1}^infty n^2,left(2,pi,n^2-3,x^2right),e^{-frac{pi,n^2}{x^2}},,quad x>0$

The $g_i(x)$ functions defined above are related to the Riemann Xi function $xi(s)$ as follows.

(17) $quadfrac{sqrt{pi}}{2},sintlimits_0^infty g_1(x),x^{-s-1},dx=pi^{frac{s+1}{2}},xi(s+1),,quadRe(s)>0$

(18) $quadfrac{1}{2},sintlimits_0^infty g_2(x),x^{-s-1},dx=xi(s+1),,qquadqquadRe(s)>0$

(19) $quadfrac{1}{2},sintlimits_0^infty g_5(x),x^{-s-1},dx=pi^{frac{s}{2}},xi(s),,qquadqquad,,Re(s)>1$

(20) $quadfrac{1}{2},sintlimits_0^infty g_6(x),x^{-s-1},dx=xi(s),,qquadqquadqquadRe(s)>1$

The following plot illustrates the $g_1(x)$, $g_2(x)$, $g_5(x)$, and $g_6(x)$ functions defined above in blue, orange, green, and red respectively where the series for each function is evaluated over the first $1,000$ terms.

Figure (1): Illustration of $g_1(x)$, $g_2(x)$, $g_5(x)$, and $g_6(x)$ (blue, orange, green, and red)

The following plot illustrates the $g_1'(x)$, $g_2'(x)$, $g_5'(x)$, and $g_6'(x)$ functions defined above in blue, orange, green, and red respectively where the series for each function is again evaluated over the first $1,000$ terms.

Figure (2): Illustration of $g_1'(x)$, $g_2'(x)$, $g_5'(x)$, and $g_6'(x)$ (blue, orange, green, and red)

Note all four of the $g_i(x)$ functions illustrated in Figure (1) above seem to have the properties of a Cumulative Distribution Function (CDF), and all four of the corresponding $g_i'(x)$ functions illustrated in Figure (2) above seem have the properties of the corresponding Probability Density Function (PDF).

Question (1): Can each of the $g_i(x)$/$g_i'(x)$ function pairs illustrated in Figures (1) and (2) above be defined in terms of some known type of probability distribution?

Question (2): When each $g_i(x)$/$g_i'(x)$ function pair is interpreted as a CDF/PDF function pair, what probability does each of these function pairs represent?

ref(1): Questions related to $f(x)$ where the Riemann Xi function $xi(s)=sintlimits_0^infty f(x),x^{-s-1},dx$

ref(2): Question on Relationship between Number Theory and Quantum Mechanics