Compute by a Monte Carlo Method 3 points Monte Carlo methods are any numerical methods determined by the use of random numbers in processes-like radioactive decay of a particle or the likelihood of a single molecule to react (Note that the physical processes aren't called "Monte Carlo", only the numerical methods using random numbers.) We will use a random method, called Buffon's Needle. to estimate the value of . Consider a circle inscribed inside of a square. If the circle has unit radius, r-1, then its area A = r2-T. The square in which it is inscribed has vertices (-1,-1), (+1,-1). (+1, +1), (-1, +1) with area A = 2 2 = 4. If you consider throwing a dart at the square, the probability that it lands inside of the circle is We can thus use nade as a proxy for Acircle and treat the square similary As are Compose a function mc-pi( n ) to estimate the value of using the Buffon's Needle method, n describes the number of points to be used in the simulation. mc_pi should return its estimate of the value of as a float . Your process should look like the following 1. Prepare an array of coordinate pairs xy. This should be of shape n,2) selected from an appropriate distribution (see notes 1 and 2 below) 2 Calculate the number of coordinate pairs inside the circle's radius. (How would you do this mathematically? Can you do this in NumPy without a loop?-although a loop is okay.) 3. Calculate the ratio eiAcirele 4. Return this estimate of 5. You may find it edifying to try the following values of n , and compare each result to the value of math. pi : 10 1 , 1ee. 1e4 1e5 , 1e6, 1e7 1e8. How does the tcirclo which implies (following the development above), 4 nsquare Asquare computational time vary? How about the accuracy of the estimate of ? You will need to consider the following notes 1. Which kind of distribution is most appropriate for randomly sampling the entire area? (Hint: if we could aim, it would be the normal distribution-but we shouldn't aim in this 2 Since numpy.random distributions accept sizes as arguments, you could use something like np.random.distribution( n,2 to generate coordinate pairs (in the range [0,1) which you'll then need to transform)-but use the right distribution! Given a distribution from [0, 1), how would you transform it to encompass the range-1,1)? (You can do this to the entire array at once since addition and multiplication are vectorized operations.) Your submission should include a function mc_pin) (This assignment is based on Langtangen, Exercise 8.29.) Compute by a Monte Carlo Method 3 points Monte Carlo methods are any numerical methods determined by the use of random numbers in processes-like radioactive decay of a particle or the likelihood of a single molecule to react (Note that the physical processes aren't called "Monte Carlo", only the numerical methods using random numbers.) We will use a random method, called Buffon's Needle. to estimate the value of . Consider a circle inscribed inside of a square. If the circle has unit radius, r-1, then its area A = r2-T. The square in which it is inscribed has vertices (-1,-1), (+1,-1). (+1, +1), (-1, +1) with area A = 2 2 = 4. If you consider throwing a dart at the square, the probability that it lands inside of the circle is We can thus use nade as a proxy for Acircle and treat the square similary As are Compose a function mc-pi( n ) to estimate the value of using the Buffon's Needle method, n describes the number of points to be used in the simulation. mc_pi should return its estimate of the value of as a float . Your process should look like the following 1. Prepare an array of coordinate pairs xy. This should be of shape n,2) selected from an appropriate distribution (see notes 1 and 2 below) 2 Calculate the number of coordinate pairs inside the circle's radius. (How would you do this mathematically? Can you do this in NumPy without a loop?-although a loop is okay.) 3. Calculate the ratio eiAcirele 4. Return this estimate of 5. You may find it edifying to try the following values of n , and compare each result to the value of math. pi : 10 1 , 1ee. 1e4 1e5 , 1e6, 1e7 1e8. How does the tcirclo which implies (following the development above), 4 nsquare Asquare computational time vary? How about the accuracy of the estimate of ? You will need to consider the following notes 1. Which kind of distribution is most appropriate for randomly sampling the entire area? (Hint: if we could aim, it would be the normal distribution-but we shouldn't aim in this 2 Since numpy.random distributions accept sizes as arguments, you could use something like np.random.distribution( n,2 to generate coordinate pairs (in the range [0,1) which you'll then need to transform)-but use the right distribution! Given a distribution from [0, 1), how would you transform it to encompass the range-1,1)? (You can do this to the entire array at once since addition and multiplication are vectorized operations.) Your submission should include a function mc_pin) (This assignment is based on Langtangen, Exercise 8.29.)
