discrete to continuous cumulative density function - c++

Question

My question is that is it possible to convert a vector which stores samples of original CDF (cumulative density function)...
something like this:
class normal
{
public:
float mean;
float sigma;
float variance = sigma * sigma;
float left_margin = mean - 4 * sigma;
float right_margin = mean + 4 * sigma;
normal():mean(0), sigma(1){}
normal(float m, float s):mean(m), sigma(s){}
float cdf(float x);
float pdf(float x);
};
float normal::pdf(float x)
{
if (x < left_margin || x > right_margin) return 0;
float coefficient = 1 / (float)sqrt(2 * PI * variance);
float x_mean = x - mean;
float result = coefficient * exp(-(x_mean * x_mean) / 2 * variance);
return result;
}
float normal::cdf(float x)
{
if (x <= left_margin) return 0;
if (x >= right_margin) return 1;
float x_mean = x - mean;
float result = (float)(0.5 * (1 + erf((x_mean) / sqrt(2 * variance))));
if (result > 1) return 1;
else return result;
}
std::vector<float> discrete_normal_cdf(normal& X)
{
std::vector<float> vec;
float L = (float)(X.left_margin);
float R = (float)(1.2 * X.right_margin);
while (L <= R)
{
vec.push_back(X.cdf(L));
L = (float)(L + 0.1);
}
std::vector<float> tmp;
// take three samples
tmp.push_back(vec.at(1)); // first non_zero element
tmp.push_back(vec.at(40)); // add element with value of 0.5
tmp.push_back(vec.at(80)); // element with value of 0.99
std::vector<float> cdf_v(5, 0);
for (auto i = 0; i < tmp.size(); i++)
cdf_v.push_back(tmp.at(i));
int l = 0;
while (l < 5)
{
cdf_v.push_back(1);
l++;
}
return cdf_v;
}
In fact what I need is this: if we have a normal
normal n1(5, 1);
take samples of its CDF to piece wise linear CDF:
vector<float> foo = discrete_normal_cdf(n1);
then reconstruct the piecewise linear CDF into normal
normal function(foo)
{
return normal(5, 1);
}
Is this function valid?
I wrote a function which takes a vector as an input
and search all the elements of the vector the for the value of 0.5
and returns the index of that element as the mean of the normal
but it not always true.
normal vec2normal(vector<float>& vec)
{
int mean;
mean = std::find(vec.begin(), vec.end(), 0.5) - vec.begin();
return normal(mean, 1);
}
I have no idea how to do this, so any suggestions will be appreciated
thank you.