Add code for the plot in double q learning problem.

Author: Makkar

.DS_Store

@@ -0,0 +1,27 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+Nmax = 1_000_000
+R = 100  # number of Monte Carlo trials (tune)
+
+sum_running_max = np.zeros(Nmax, dtype=np.float64)
+
+rng = np.random.default_rng(0)
+for r in range(R):
+    eps = rng.standard_normal(Nmax)          # N(0,1)
+    runmax = np.maximum.accumulate(eps)      # runmax[n-1] = max_{i<=n} eps[i]
+    sum_running_max += runmax
+
+fhat = 1.0 + (sum_running_max / R)
+ghat = np.ones(Nmax)  # exactly g(n)=1, but included for plotting
+
+n = np.arange(1, Nmax + 1)
+
+plt.figure()
+plt.semilogx(n, fhat, label="Monte Carlo  f(n)=E[max X]")
+plt.semilogx(n, ghat, label="g(n)=max E[X]=1")
+plt.xlabel("n (log scale)")
+plt.ylabel("value")
+plt.legend()
+plt.grid(True, which="both", linestyle="--", linewidth=0.5)
+plt.show()