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A family of such functions is given by $$\phi(x)=\phi_c(x):=e^{t_cx+cx^2}$$ for all real $x$, where $c\in(-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2)$ and $t_c=\pm\sqrt{1 - 8 c + 12 c^2 + 16 c^3}$; then both expected values equal ${e^{{2 t_c^2}/(1-4 c)}}/{\sqrt{1-4 c}}={e^{{2 - 8 c - 8 c^2}}}/{\sqrt{1-4 c}}$.

In particular, choosing here $c=0$, we get two members of this family, given by $\phi_0(x)=e^{\pm x}$ for all real $x$.

A family of such functions is given by $$\phi(x)=\phi_c(x):=e^{t_cx+cx^2}$$ for all real $x$, where $c\in[-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2]$ and $t_c=\pm\sqrt{1 - 8 c + 12 c^2 + 16 c^3}$; then both expected values equal ${e^{{2 t_c^2}/(1-4 c)}}/{\sqrt{1-4 c}}={e^{{2 - 8 c - 8 c^2}}}/{\sqrt{1-4 c}}$.

In particular, choosing here $c=0$, we get two members of this family, given by $\phi_0(x)=e^{\pm x}$ for all real $x$.

Another two members of this family are given by $\phi_c(x)=e^{cx^2}$ for $c\in\{-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2\}$ and all real $x$.

A family of such functions is given by $$\phi(x)=\phi_c(x):=e^{t_cx+cx^2}$$ for all real $x$, where $c\in(-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2)$ and $t_c=\pm\sqrt{1 - 8 c + 12 c^2 + 16 c^3}$; then both expected values equal ${e^{{2 t^2}/(1-4 c)}}/{\sqrt{1-4 c}}$.

In particular, choosing here $c=0$, we get two members of this family, given by $\phi_0(x)=e^{\pm x}$ for all real $x$.

A family of such functions is given by $$\phi(x)=\phi_c(x):=e^{t_cx+cx^2}$$ for all real $x$, where $c\in(-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2)$ and $t_c=\pm\sqrt{1 - 8 c + 12 c^2 + 16 c^3}$; then both expected values equal ${e^{{2 t_c^2}/(1-4 c)}}/{\sqrt{1-4 c}}={e^{{2 - 8 c - 8 c^2}}}/{\sqrt{1-4 c}}$.

In particular, choosing here $c=0$, we get two members of this family, given by $\phi_0(x)=e^{\pm x}$ for all real $x$.

A family of such functions is given by $$\phi(x)=\phi_c(x):=e^{t_cx+cx^2}$$ for all real $x$, where $c\in(-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2)$ and $t_c=\pm\sqrt{1 - 8 c + 12 c^2 + 16 c^3}$; then both expected values equal ${e^{{2 t^2}/(1-4 c)}}/{\sqrt{1-4 c}}$.

A family of such functions is given by $$\phi(x)=\phi_c(x):=e^{t_cx+cx^2}$$ for all real $x$, where $c\in(-\frac{1+\sqrt2}2,\frac{-1+\sqrt2}2)$ and $t_c=\pm\sqrt{1 - 8 c + 12 c^2 + 16 c^3}$; then both expected values equal ${e^{{2 t^2}/(1-4 c)}}/{\sqrt{1-4 c}}$.

In particular, choosing here $c=0$, we get two members of this family, given by $\phi_0(x)=e^{\pm x}$ for all real $x$.