Cannot Plot Function

Question

I cannot plot this function as I am getting errors. I am not sure what I am doing wrong.

Seems to work when $$n \rightarrow 1,$$ but when $$n \rightarrow 2,$$ I get a whole bunch of errors and nothing is plotted.

Here is the code:

$$ \text{Plot}\left[\frac{n e^{-t (\lambda +\mu )} I_1\left(2 t \sqrt{\lambda \mu }\right) \left(\int_0^t \frac{e^{t (-\lambda -\mu )} I_1\left(2 t \sqrt{\lambda \mu }\right)}{\sqrt{\rho } t} \, dt\right){}^{n-1}}{\sqrt{\rho } t}\text{/.}\, \left\{\lambda \to 0.99,\mu \to 1,\rho \to \frac{1}{2},n\to \{2\}\right\},\{t,0,10\},\text{PlotRange}\to \text{All}\right]$$

Here is copyable code:

Plot[(n*BesselI[1, 2*t*Sqrt[λ*μ]]*            
      Integrate[(E^(t*(-λ - μ))*
           BesselI[1, 2*t*Sqrt[λ*μ]])/(t*Sqrt[ρ]), 
                {t, 0, t}]^(-1 + n))/
    E^(t*(λ + μ))/(t*Sqrt[ρ]) /. 
     {λ -> 0.99, μ -> 1, ρ -> 1/2, n -> {2}}, {t, 0., 10}, PlotRange -> All]

Answer 1

fun[t_, n_, \[Rho]_, \[Lambda]_, \[Mu]_] := (n*
     BesselI[1, 2*t*Sqrt[\[Lambda]*\[Mu]]]*
     NIntegrate[(E^(s*(-\[Lambda] - \[Mu]))*
          BesselI[1, 2*s*Sqrt[\[Lambda]*\[Mu]]])/(s*Sqrt[\[Rho]]), {s,
         0, t}]^(-1 + n))/E^(t*(\[Lambda] + \[Mu]))/(t*Sqrt[\[Rho]])

Visualizing:

Manipulate[
 Plot[fun[t, j, 1/2, 0.99, 1], {t, 0, 10}, 
  PlotRange -> {0, 1}], {{j, 0.5, "n"}, 0.5, 2.5, 
  Appearance -> "Labeled"}]

Answer 2

Is this how the function supposed to look like?

 int[t1_, lam_, mu_, rao_] := NIntegrate[(E^(t*(-lam - mu))*
      BesselI[1, 2*t*Sqrt[lam*mu]])/(t*Sqrt[rao]), {t, 0, t1}];

f[t_, {lam_, mu_, rao_, n_}] := (n*BesselI[1, 2*t*Sqrt[lam*mu]]*
   int[t, lam, mu, rao]^(-1 + n))/E^(t*(lam + mu))/(t*Sqrt[rao]);

parm = {lam -> 0.99, mu -> 1, rao -> 1/2, n -> 2};
Plot[f[t, {lam, mu, rao, n} /. parm], {t, 0, 10}, PlotRange -> All, 
   Frame -> True, GridLines -> Automatic,  GridLinesStyle -> LightGray]