#!/usr/bin/env python
# coding: utf-8

# # [Homework Week 6](https://tsoo-math.github.io/ucl2/2021-HW-week6.html)
# 
# For complete written solutions, please [see](https://tsoo-math.github.io/ucl/QHW1-sol.html).  We will mainly be concerned with the Python code here.

# ## Integration

# In[1]:


import numpy as np

def g(x):                            # defines the function g
    y=np.power(x,2) * np.sin(x)
    return y 

z = np.random.exponential(scale=1.0, size=10000)   # simulates 10000 iid exp
integralapprox = np.average(g(z))        # approximation using the law of large numbers
print(integralapprox)
    


# ## Grouping coins

# In[2]:


def isFour(x):  # function which checks if the binary sequence has a run of four heads
    ans =0
    for i in range(len(x) +1-4 ):
        if (x[i]==1 and x[i+1]==1 and x[i+2]==1 and x[i+3]==1):
            ans=1
    return ans

def check():   # checking if a single random binary sequence of length 20 has a run of four heads
    binseq = np.random.binomial(1, 0.5, 20)
    return isFour(binseq)

num = [check()  for _ in range(10000)]  # run the check function 10000 times
print(np.average(num))


# ## Pen and paper

# In[3]:


def T():
    s=0
    n=0
    while s<1:
        s = s + np.random.uniform()
        n=n+1
    return n

num = [T()  for _ in range(100000)]
print(np.average(num)) 


# ## Endnotes
# 
# Use the ipynb [source](https://tsoo-math.github.io/ucl2/2021-hw-week6.ipynb) for the most update version.

# In[4]:


from datetime import datetime
print(datetime.now())