Suppose I have a PDF with mean a and standard deviation b defined as below. As the value of k increases, a should decrease, while b should also decrease. This should cause a decrease in the output, but for some reason, there is an increase. Even when I set the value of b constant, and simply let a decrease, the output increases, which should not happen. Any thoughts?

a=1/2Erfc[k/Sqrt[2]] * x
b= -q Log[2, q] - (1 - q) Log[2, 1 - q]


1 Answer 1


This is really just a math problem. Mathematica is providing the correct output based on your input.

The output you're asking for (in shorthand) is x/((CDF[8]-CDF1)/(CDF[8]-CDF1)). This can be rewritten as (CDF[8]-CDF[0]) * x / (CDF[8]-CDF1). The numerator (CDF[8]-CDF[0]) goes somewhat slowly to 1/2 as k increases. The denominator (CDF[8]-CDF1) rapidly goes to zero. When you have a small number in the denominator and a much larger number in the numerator, it explodes to infinity.

As k goes to zero, your output should converge to 7.0421, since the numerator and denominator are basically the same.

Here is some code to see what happens:

x = 7;
a[k_] := 1/2 Erfc[k/Sqrt[2]]*x
q[k_] := a[k]/x
b[k_] := -q[k] Log[2, q[k]] - (1 - q[k]) Log[2, 1 - q[k]]
cdf[val_, k_] := CDF[NormalDistribution[a[k], b[k]], val]

(* Let's see what happens to your numerator and denominator *)
 {cdf[8, k] - cdf[0, k],
  cdf[8, k] - cdf[1, k]},
 {k, 0, 3},
 AxesLabel -> {"k", "Value"},
 PlotLegends -> 
    "Denominator"}, {Scaled[{0.95, 0.25}], {1, 0}}]]

(* Let's see what happens to the mean and standard deviation as k increases *)
 {a[k], b[k]},
 {k, 0, 3},
 AxesLabel -> {"k", "Value"},
 PlotLegends -> 
  Placed[{"Mean (a)", 
    "Standard Deviation (b)"}, {Scaled[{0.9, 0.9}], {1, 1}}],
 PlotRange -> Full]

You should see that the denominator (yellow curve) rapidly drops to zero, meaning your output value will explode to infinity.

Plot of numerator and denominator.

  • $\begingroup$ Yes, you are correct. I have realized that I am asking for the wrong output. Thank you for helping! $\endgroup$
    – sde7
    Commented Nov 24, 2018 at 18:33
  • $\begingroup$ No problem, glad it worked out for you! $\endgroup$
    – MassDefect
    Commented Nov 25, 2018 at 1:56

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