Pnoise[V_] = 
      1/(Sqrt[2 Pi] Vn)  Exp[-((V/Vn)^2/
                2)]

(∫_(-∞)^∞) Pnoise[
          V] \[DifferentialD]V

(∫_(-∞)^∞) V^2 Pnoise[
          V] \[DifferentialD]V

<< Graphics`Legend`

matlabplotstyle = {{Hue[0.75], Dashing[{0.05, 0.05}]}, {Hue[ .25], 
          Dashing[{0.05, 0.02}]}, {Hue[1], Dashing[{0.1, 0.05}]}, {Hue[0.5], 
          Dashing[{0.13, 0.02}]}}

\(Plot[
      Pnoise[V] /. Vn -> 1, {V, \(-5\), 10}, {PlotRange -> All, 
        AxesLabel -> {V, \*""\!\(P\_noise\)(V)""}}]\)

(∫_t^∞) Pnoise[V] \[DifferentialD]V

(1/((Sqrt[2 π]) Vn)) (Sqrt[(π/2)]) (Sqrt[(Vn^2)]) Erfc[
          (t/((Sqrt[2]) (Sqrt[(Vn^2)])))] //. {Sqrt[Vn^2] -> Vn, 
        1/Sqrt[Vn^2] -> 1/Vn}

Pfa = (1/2) Erfc[(t/((Sqrt[2]) Vn))] /. Vn -> 1

Plot[Pfa, {t, -5, 10},  
      AxesLabel -> {"t, Threshold", (Subscript[P, d])}]

Psignal[V_] = 
      1/(Sqrt[2 Pi] Var)  Exp[-(((V - Vs)/Var)^2/
                2)]

(∫_t^∞) Psignal[V] \[DifferentialD]V

Plot[{Pnoise[V] /. Vn -> 1, 
        Psignal[V] //. {Vn -> 1, vsnr -> 2, 
            Var -> Sqrt[Vn^2 + vsnr^2/100], 
            Vs -> vsnr}}, {V, -5, 10}, {PlotRange -> All, 
        AxesLabel -> {V, "P(V)"}, 
        PlotStyle -> matlabplotstyle}]

Plot[{Pnoise[V] /. Vn -> 1, 
        Psignal[V] //. {Vn -> 1, vsnr -> 3, 
            Var -> Sqrt[Vn^2 + vsnr^2/100], 
            Vs -> vsnr}}, {V, -5, 10}, {PlotRange -> All, 
        AxesLabel -> {V, "P(V)"}, 
        PlotStyle -> matlabplotstyle}]

Pdet[speckles_, 
        vsnr_] = (1/((Sqrt[2 π]) Var)) (Sqrt[ (π/2)]) (Sqrt[(Var^2)]) Erfc[((t - Vs)/((Sqrt[2]) (Sqrt[(Var^2)])))] //. {Vn -> 1, 
          Var -> Sqrt[Vn^2 + vsnr^2/speckles], 
          Vs -> vsnr}

\(Plot[{Pfa, \((Pdet[100, 2])\), Pdet[100, 3]}, {t, \(-5\), 
        10}, {AxesLabel -> {"t, Threshold", \*""Probability, (Subscript[P, d]),(Subscript[P, fa])""}, PlotStyle -> matlabplotstyle}]\)

\(ParametricPlot[{{Pfa, \((Pdet[100, 2])\)}, {Pfa, \((Pdet[100, 
              3])\)}}, {t, \(-5\), 
        10}, {AxesLabel -> {\*""\!\(P\_fa\)"", \*""\!\(P\_d\)""}, 
        PlotStyle -> matlabplotstyle}]\)

Pcnoise[p_] = (1/pn) Exp[-p/pn]

(∫_0^∞) Pcnoise[p] \[DifferentialD]p

(∫_0^∞) p Pcnoise[p] \[DifferentialD]p

Pcsignal[p_] = (1/(pn + ps)) 
        Exp[-p/(pn + ps)]

Plot[{Pcnoise[p] /. pn -> 1, 
        Pcsignal[p] //. {pn -> 1, ps -> 4 pn}}, {p, 0, 
        10}, {PlotRange -> All, 
        AxesLabel -> {"P, Power", "P(P), Probability Density"}, 
        PlotStyle -> matlabplotstyle}]

Pcfa = (∫_t^∞)(Pcnoise[p] /. 
              pn -> 1) \[DifferentialD]p

(∫_t^∞)(Pcsignal[p] /. 
            pn -> 1) \[DifferentialD]p

Pcdet[ps_] = (E^(-(t/(1 + ps))))

\(Plot[{Pcfa, Pcdet[4]}, {t, 0, 
        10}, {AxesLabel -> {"t, Threshold", \*""(Subscript[P, fa]), (Subscript[P, det])""}, PlotStyle -> matlabplotstyle}]\)

\(ParametricPlot[{{Pcfa, Pcdet[4]}, {Pcfa, Pcdet[9]}}, {t, 0, 
        10}, {AxesLabel -> {\*""(Subscript[P, fa])"", \*""(Subscript[P, det])""}, PlotStyle -> matlabplotstyle}]\)

\(ParametricPlot[{{Pfa, Pdet[100, 2]}, {Pcfa, Pcdet[4]}, {Pfa, 
          Pdet[100, 3]}, {Pcfa, Pcdet[9]}}, {t, 0, 
        10}, {AxesLabel -> {\*""\!\(P\_fa\)"", \*""\!\(P\_d\)""}, 
        PlotStyle -> {{Hue[0.75], Dashing[{0.05, 0.05}]}, {Hue[ .25], 
              Dashing[{0.05, 0.02}]}, {Hue[1], 
              Dashing[{0.1, 0.05}]}, {Hue[0.5], 
              Dashing[{0.13, 0.02}]}}}]\)