Raising the normal distribution curve to a specified height on the $y$-axis

Question

I'd like to know how to fit the normal distribution curve between $x$ and $y$ values so that the curve maintains its bell shape. I guessed the sigma is the scale ratio, but it doesn't work as I suspected.

The following is my Mathematica code where I tried to get the maximum to be $y = 15$ at $x = 0$. I also want $x$ to range over $0,\, 1,\,\ldots, 100$. I have started the plot from -50 to get the whole shape of the curve.

With[{sigma = 15, mu = 0},
  t = Table[E^(-(x - mu)^2/(2 sigma^2)), {x, -50, 50}];
  ListPlot[t #& @ #[t]& /@ {Mean, Variance, Median},  
  PlotRange -> Full, Filling -> Bottom]]

I thought that I could use median, or variance, or mean values to lift the curve up to 15, but it didn't work as I expected.

Can you help modify the code so that the curve will reach $y = 15$ rather than $y \sim 0.38$ and still keep the shape of a normal distribution curve?

Below is the image of the plots from the code above.

Answer 1

With[{sigma = 15, mu = 0}, 
 t = Table[E^(-(x - mu)^2/(2 sigma^2)), {x, -50, 50}];
 ListPlot[# Rescale[t] & /@ {5, 10, 15}, PlotRange -> Full, 
  Filling -> Bottom, DataRange -> {-50, 50}]]

Alternatively,

With[{sigma = 15, mu = 0}, 
 t = Table[{x, E^(-(x - mu)^2/(2 sigma^2))}, {x, -50, 50}];
 ListPlot[ScalingTransform[{1, #}][t] & /@ {5, 10, 15}, 
  PlotRange -> All, Filling -> Bottom ]]