Gain insights into Python for scientific computing. Explore arrays, plotting, linear equations, and algorithms using NumPy, Matplotlib, SciPy. Delve into applying tools with practical exercises.

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If you're a scientist or an engineer interested in learning scientific computing, this is the place to start.

In this course, you'll learn to write your own useful code to perform impactful scientific computations. Along the way, your understanding will be tested with periodic quizzes and exercises.

Topics covered in this course include arrays, plotting, linear equations, symbolic computation, scientific algorithms, and random variables. You’ll also be exposed to popular Python packages like NumPy, Matplotlib, SciPy, and others. In the last part of the course, the application section will test your ability to recall and apply the tools you have studied into newly learned scientific concepts.

At the end of this course, you'll be equipped with the tools necessary for everyday scientific computation.

Python for Scientists and Engineers

In the example below, we will flip a coin five times in a row and record how many times we obtain tails (varying from 0-5). We will be performing the experiment 1000 times. 

We will then compute the cumulative probability, print the values to the screen and make a plot of the cumulative probability function using a bar graph.


import numpy as np
import matplotlib.pyplot as plt
import numpy.random as rnd

N = 1000
tails = np.sum(rnd.randint(0, 1+1, (5, 1000)), axis=0)
counttails = np.zeros(6, dtype='int')

for i in range(6):
    counttails[i] = np.count_nonzero(tails == i)

prob = counttails / N
cum_prob = np.cumsum(prob)
print('probababilties:', prob)
print('cumulative probabilities:', cum_prob)
plt.bar(range(0, 6), cum_prob)
plt.xticks(range(0, 6))
plt.xlabel('number of tails in two flips')
plt.ylabel('cumulative probability');

> <img src="/api/collection/10370001/6066104977850368/page/4580102618742784/image/6441996354846720" width="35"> Execute the code several times and see that the graph changes a bit every time.

# Probability of a Bernoulli variable

In the example above, we computed the probability of a certain number of heads in five flips experimentally. But we can, of course, compute the value exactly by using a few simple formulas. 

Consider the random variable $Y$, which is the outcome of an experiment with two possible values $0$ and $1$. Let $p$ be the probability of success, 

$$p=P(Y=1)$$ 

Then $Y$ is said to be a Bernoulli variable. The experiment is repeated $n$ times and we define $X$ as the number of successes in the experiment. The variable $X$ has a Binomial Distribution with parameters $n$ and $p$. The probability that $X$ takes value $k$ can be computed as

$$P(X=k)=\left(\begin{array}{}n\\ k\end{array}\right)p^k(1-p)^{n-k}$$

The term $\left(\begin{array}{}n\\ k\end{array}\right)$ may be computed with the `comb()` function, which needs to be imported from the `scipy.special` package.


Let's go back to the experiment where we flip a coin five times in a row and record how many times we obtain tails. We will compute the theoretical probability for 0, 1, 2, 3, 4, and 5 tails and compare the answer to the probability computed from 1000 trials, 10000 trials, and 100000 trials. Let' see if we can approach the theoretical value with more trials?

In this lesson, we will learn about finding the probability of a Bernoulli variable. 

Introduction

Python Refresher

Arrays

Plotting

Systems of Linear Equations

Symbolic Computation

Scientific Algorithms

Random Variables

Applications

Conclusion

Appendix

Python for Scientists and Engineers Exam

Bernoulli Variable

Probability of a