To determine if there exist rational numbers ( q_1 \in \mathbb{Q}^* ) (non-zero) and ( q_2 \in \mathbb{Q} ) such that ( \Phi(q_1) = q_2 ), where ( \Phi ) is the standard normal cumulative distribution function, we analyze the properties of ( \Phi ) and the error function ( \text{erf} ).

Key Observations:

1. Definition of ( \Phi ): The standard normal CDF is given by ( \Phi(...

Short answer: This is not known. It is widely believed that the only rational value of the standard normal CDF at a rational point is Φ(0) = 1/2, but no proof is known.

Details:

• Your condition Φ(q1) = q2 with q1, q2 ∈ Q is equivalent to asking whether the error function erf takes a rational value at a rational point, since Φ(x) = 1/2(1 + erf(x/√2)). So Φ(q1) ∈ Q is equivalent to erf(q1/√2) ...

Write Φ(x) for the standard‐normal CDF

 Φ(x) = (2π)^(–1/2) ∫_{–∞}^x e^{–t²/2} dt.

We are asking:

“If q₁,q₂∈ℚ satisfy Φ(q₁)=q₂, must it be that q₁=0 and q₂=½?”

Remark. Φ can be written in terms of the “error‐function”
 erf z = (2/√π)∫₀^z e^{–t²} dt
via
 Φ(x) = ½ [1+erf (x/√2)].

Thus the question is equivalent to asking whether erf (α) can be rational for any nonzero rational α. ...

Let

  Φ(x)= 1/√(2π)⋅∫_{-∞}^{x} e^{-t²/2},dt ,  x∈ℝ ,

and assume that q₁ , q₂ are rational numbers with

  Φ(q₁)=q₂. (1)

Question: must we have q₁=0 and q₂=1/2 ?

What is known

1. Relation with the error function
     Φ(x)=½(1+erf(x/√2)),  erf(z)= 2/√π ∫₀^{z} e^{-t²},dt .
     Hence (1) is equivalent to

  erf(α)=2q₂−1  with  α = q₁/√2 . ...

<think> The problem asks if the standard normal cumulative distribution function, denoted as Φ(q), can be rational for some rational q not equal to zero. Specifically, it gives a scenario where q1 and q2 are rational numbers such that the integral from negative infinity to q1 of the standard normal density equals q2. The question is whether this forces q1 to be zero and q2 to be one-half.

First, ...

This is an excellent and profound question that delves into the fascinating field of transcendental number theory.

Summary of the Answer

The short answer is: This is a famous open problem. It is widely conjectured that the only point on the graph of the standard normal CDF, , with rational coordinates is , but this has not been proven.

Therefore, we cannot definitivel...

Is rational for some , where is the standard normal cumulative distribution function?

This question asks whether there are any rational numbers and such that , where is the standard normal CDF.

Let me approach this systematically:

Reformulating the problem

The standard normal CDF is defined as...

This is a fascinating question about the transcendence properties of the normal distribution function. Let me analyze this carefully.

The question asks whether for rational numbers implies that necessarily and .

First, let's verify that is indeed a solution:

Let

  Φ(x)= 1/√(2π) ∫_{-∞}^{x} e^{-t²/2} dt (x∈ℝ)

denote the standard normal cumulative distribution function.
Put q₁∈ℚ*, q₂∈ℚ and assume

  Φ(q₁)=q₂. (1)

The question is whether (1) forces the “obvious’’ pair

  (q₁ , q₂)=(0 , 1/2).

Short answer. No proof of such a statement is known. In fact we do not even know ...