# compose (non-)symbolic functions

I wonder how to compose with a non-symbolic function, or, more generally, how to compose several functions, if some of them are non-symbolic.

In my concrete case, I wish to plot a transformed Gaussian distribution function (concretely, the Camp-Paulson approximation), comparing with a plot of probabilities of the binomial distribution this approximates.

What works, but is not flexible enough is:

```
import scipy.stats
n=20
p=0.05
binom_dist = scipy.stats.binom(n,p)
Bin=bar_chart([binom_dist.pmf(x) for x in range(n+1)],width=1)
T = RealDistribution('gaussian', 1)
CP=plot(lambda k:T.distribution_function(-1/3*(((0.95*k + 0.95)/(-0.05*k + 1.0))^(1/3)*(1/(k + 1) - 9) + 1/(k - 20) + 9)/sqrt(((0.95*k + 0.95)/(-0.05*k + 1.0))^(2/3)/(k + 1) - 1/(k - 20))),(k,-1,21) , rgbcolor=(0.8,0,0))
show(Bin+CP)
```

This option has the argument of T.distribution_function hard coded. I rather would have this computed from n and p, using another function. I have two approaches so far, that both don't work.

The first one has a function K computing the input to T.distribution_function like so:

```
import scipy.stats
n=20
p=0.05
binom_dist = scipy.stats.binom(n,p)
Bin=bar_chart([binom_dist.pmf(x) for x in range(n+1)],width=1)
T = RealDistribution('gaussian', 1)
#
a = 1/(9*n-9*k)
b = 1/(9*k+9)
r = (k+1)*(1-p)/(n*p-k*p)
c = (1-b)*r^(1/3)
μ = 1 - a
σ = sqrt(b*r^(2/3) + a)
def K(k): return (c - μ)/σ
#
CP=plot(lambda k:T.distribution_function(K(k)),(k,-1,21), rgbcolor=(0.8,0,0))
show(Bin+CP)
```

another alternative tries to set up the composed function before entering plot like so:

```
import scipy.stats
n=20
p=0.05
binom_dist = scipy.stats.binom(n,p)
Bin=bar_chart([binom_dist.pmf(x) for x in range(n+1)],width=1)
T = RealDistribution('gaussian', 1)
#
a = 1/(9*n-9*k)
b = 1/(9*k+9)
r = (k+1)*(1-p)/(n*p-k*p)
c = (1-b)*r^(1/3)
μ = 1 - a
σ = sqrt(b*r^(2/3) + a)
def distCPapprox(k): return T.distribution_function((c - μ)/σ)
#
CP=plot(lambda k:distCPapprox(k),(k,-1,21), rgbcolor=(0.8,0,0))
show(Bin+CP)
```

Both approaches don't work. I searched extensively for advice for function composition but couldn't find anything helpful. Your advice is greatly appreciated.

The following idea seems to go in the correct direction:

And, finally, what I intended seems to be the following. (However, I still would like to know how composition of functions, some of which are non-symbolic works.)