solution

Let {Yn, n ≥ 1} be i.i.d. U [0, 1] random variables and {Zn, n ≥ 1}
be i.i.d. N(0, 1) random variables. Define Xn = 1{Yn ≥ a}+Zn for some constant
a. The goal of the problem is to design an algorithm that “learns” the value of a
from the observation of pairs (Xn, Y n). We construct a model

with λ = 10. Note that when u > 0, the denominator of g(u) is close to 1, so
that g(u) ≈ 1. Also, when u ≈ 0. Thus,
g(u) ≈ 1{u ≥ 0}. The function g(·) is called the logistic function. Use SGD in
Python to estimate θ (Fig. 12.17).