PJ3

Nhat Cao16.90 Project 3

Probabilistic Simulation of a Baseball Batter

I. Implementation of the Monte Carlo simulationTo implement the Monte Carlo method, the given codes for random number generators of different distributions were modified, using the definitions from the Course Notes. The generators are fairly simple and only the functions rand and randn were needed. For the main Monte Carlo code, parameters of different distributions for the components of bb_dyn the trajectory calculator, are input. Then, for a specified number of iterations, parameters are drawn from their distribution and the trajectory is calculated. The batted trajectory is then categorized and recorded. The history of the input parameters, range, and height is recorded for histogramsThe standard error in estimating probabilities is also calculated, and the number of iterations is adjusted upon this information. If the number of iterations is not large enough, it will be increased such that the error of the desired probability is smaller than 0.003, satisfying the 0.01 envelope at 99% confidence.II. Test driveTo make sure the sample schemes are working as intended, the following histograms are generated from the input distributions

Fig 1. Normal Distribution of Wind Speed

Fig 2. Log normal distribution of Bat Speed

Fig 3. Triangular distribution of Spin

Fig 4. Triangular Distribution of Cda

III. ProbabilitiesThe following probabilities are generated for all there pitch types. The iteration size is increased until all probabilities meet the required confidence envelope, meaning the calculated standard error of each is less than 0.0033.Pitch typeHigh FastballSinking FastballCurveball

Ground Ball00.54090.1383

Line Drive00.29800.0663

Fly Ball0.71530.16050.3492

Home Run0.28470.00070.4462

Iterations200002200025000

IV. Histograms of Range and inputsCase 1: High Fastball with 20000 iterations

Case 2: Sinking Fastball with 22000 iterations

Case 3: Curveball with 25000 iterations

The histograms do not vary much for Vwind, Vbat, Cda, Cdb and Cm because they are the same distributions for all three pitches. The distributions for Vball are all log-normal, but the different variance make the spread envelopes different between pitches. The triangular distributions for theta and omega also have different spreads and range of value.

V. IntepretationThe sinking fastball gives the highest probability of ground ball hits, so I would recommend it to the pitcher to create a double-play. On the other hand, the batters should probably hit a curveball because it has the highest probability of home run hits.

VI. Estimation of the Mean DistanceThe following estimations are made for the different pitchesPitch TypeHigh FastballSinking FastballCurveball

Mean357.968.6345.8

Variance5477376720473

Standard Error0.27390.17120.8189