straw/pm21-dragon

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Andrew Straw 50f3122dee lecture 12 as finished

2025-01-24 08:16:44 +01:00

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Learning goals¶

Understand the following:

Probability distribution
Probability density function (PDF)
Gaussian distribution
Histogram
Mean
Standard deviation
Variance
Mode

We will discuss these as we work through this example

In [1]:

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats

In [17]:

sample_size = 30

true_mean = 10.0
true_variance = 4.0
true_std = np.sqrt(true_variance)

rng = np.random.RandomState(123)
fish_sample = rng.normal(loc=true_mean, scale=true_std, size=sample_size)

In [8]:

fish_sample

Out[8]:

array([ 7.82873879, 11.99469089, 10.565957  ,  6.98741057,  8.8427995 ,
       13.30287307,  5.14664151,  9.14217474, 12.53187252,  8.2665192 ,
        8.6422277 ,  9.81058206, 12.98277925,  8.72219601,  9.11203608,
        9.13129745, 14.41186017, 14.37357218, 12.0081078 , 10.7723728 ,
       11.47473715, 12.98146406,  8.12833226, 12.35165809,  7.49223866,
        8.724497  , 11.81421039,  7.1426386 ,  9.71986256,  8.27649021])

In [9]:

x=plt.hist(fish_sample)
plt.xlabel("fish length (cm)")
plt.ylabel("counts");

In [10]:

Out[10]:

(array([1., 1., 3., 7., 4., 2., 2., 5., 3., 2.]),
 array([ 5.14664151,  6.07316338,  6.99968524,  7.92620711,  8.85272897,
         9.77925084, 10.7057727 , 11.63229457, 12.55881644, 13.4853383 ,
        14.41186017]),
 <BarContainer object of 10 artists>)

What is the mean of the sample of fish lengths?¶

In [11]:

sample_mean = np.mean(fish_sample)
print(sample_mean)

10.089427941923741

What is the mode) of the sample of fish lengths? ("modus)" in German.)¶

What is the sample variance of fish lengths?¶

We put the result in the variable length_variance. We use the ddof=1 keyword argument to np.var() because we are estimating variance from a sample here.

In [18]:

sample_variance = np.var(fish_sample, ddof=1)
print(sample_variance)

5.637780974408914

Probability Density Functions (PDFs)¶

for working with continuous variables (vs. probability mass functions for discrete variables)
here, the area under the curve gives the probability (in contrast to probability mass functions where we have probabilities for every single value)
the area under the whole curve is 1. In other words, some outcome must happen.

Normal Distribution (Gaussian Distribution)¶

unimodal and symmetric
many algorithms in machine learning & statistics have normality assumptions
two parameters: mean (center of the peak) and standard deviation (spread); $N(\mu, \sigma)$
we can estimate parameters $\mu$ and $\sigma$ by the sample mean ($\bar{x})$ and sample variance ($s^2$)
univariate Normal distribution:

$$N(\mu, \sigma) = f(x \mid \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \; \exp\bigg(-\frac{(x-\mu)^2}{2\sigma^2}\bigg)$$

standard normal distribution with zero mean and unit variance, $N(0, 1)$:

In [19]:

def univariate_gaussian_pdf(x, mean, variance):
    return (1. / np.sqrt(2*np.pi*variance) * 
            np.exp(- ((x - mean)**2 / (2.*variance))))

In [20]:

mean = 0
stddev = 1
x = np.arange(-5, 5, 0.01)
y = univariate_gaussian_pdf(x, mean, stddev**2)
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('Probability Density Function (PDF)')
plt.show()

In [21]:

mean = 10
stddev = 2.0
x = np.arange(5, 15, 0.01)
y = univariate_gaussian_pdf(x, mean, stddev**2)
plt.plot(x, y)
plt.xlabel('fish length (cm)')
plt.ylabel('Probability Density Function (PDF)')
plt.show()

In [22]:

plt.hist(fish_sample, density=True, label='samples')

x = np.linspace(5.0, 15.0, 100)
y = univariate_gaussian_pdf(x, true_mean, true_variance)
plt.plot(x, y, lw=5, label='theoretical (mean=%.1f, variance=%.1f)'%(true_mean,true_variance))

sample_mean = np.mean(fish_sample)
sample_variance = np.var(fish_sample, ddof=1) # for a sample variance, always use ddof=1. Here is one explanation why https://mortada.net/computing-sample-variance-why-divide-by-n-1.html
y2 = univariate_gaussian_pdf(x, sample_mean, sample_variance)
plt.plot(x, y2, label='empirical (mean=%.1f, variance=%.1f)'%(sample_mean,sample_variance))

plt.xlabel("length (cm)")
plt.ylabel("probability");
plt.legend();
# plt.savefig('fish-gaussian.png');

Discussion points¶

Here are a few questions you should be able to answer:

what is plotted on the x axis?
what is plotted on the y axis?
what is the area under the curve from -∞ to ∞?
according to our Guassian distribution, what is the most likely fish length?
according to our histogram, what is the most likely fish length?

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