[prob_dist] Bernoulli distribution section - editorial suggestions

[longye-tian]

I've noticed that the content in the Bernoulli distribution section closely mirrors that of the uniform distribution section.

With this in mind, I'd like to propose the following amendments. Would that be alright with you, John(@jstac)?

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Another useful distribution is the Bernoulli distribution on $S = {0, 1}$, which has PMF:

$$p(x_i) = \begin{cases} p & \text{if $x_i=1$}\\ 1-p & \text{if $x_i = 0$}\\ \end{cases}$$

Here $x_i\in S$ is the outcome of the random variable.

The interpretation of $p(x_i)$ is the probability of 'true' for any single experiment that asks 'True-False' question.

We can import the Bernoulli distribution on $S = {0,1}$ from SciPy like so:

p = 0.4 # The probability of True

u = scipy.stats.bernoulli(p)

Here's the mean and variance:

u.mean(), u.var()

The formula for the mean is $p$, and the formula for the variance is $p(1-p)$.

Now let's evaluate the PMF:

u.pmf(0)

u.pmf(1)

Here's a plot of the probability mass function:

fig, ax = plt.subplots()
S = np.arange(-4, 6)
ax.plot(S, u.pmf(S), linestyle='', marker='o', alpha=0.8, ms=4)
ax.vlines(S, 0, u.pmf(S), lw=0.2)
ax.set_xticks(S)
plt.show()

Here's a plot of the CDF:

fig, ax = plt.subplots()
S = np.arange(-4, 6)
ax.step(S, u.cdf(S))
ax.vlines(S, 0, u.cdf(S), lw=0.2)
ax.set_xticks(S)
plt.show()

The CDF jumps $p(x_i)$ at $x_i$.

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Best,
Longye

[jstac]

Many thanks for picking this up @longye-tian.

Let's simplify this section, since the random variable is so simple. I suggest we drop the sentence "The interpretation of $p(x_i)$ is the probability of 'true' for any single experiment that asks 'True-False' question." and also the two plots.

[longye-tian]

Thank you, John. I agree that given the simplicity of the random variable, removing the specified sentence and the two plots will streamline our content effectively. I'll proceed with making these changes.

Best,
Longye