Probabilit y densit y f u nctions STATISTIC AL TH IN K IN G IN P - - PowerPoint PPT Presentation

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Probabilit y densit y f u nctions STATISTIC AL TH IN K IN G IN P - - PowerPoint PPT Presentation

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Probabilit y densit y f u nctions STATISTIC AL TH IN K IN G IN P YTH ON ( PAR T 1 ) J u stin Bois Lect u rer at the California Instit u te of Technolog y Contin u o u s v ariables Q u antities that can take an y v al u e , not j u st discrete v

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Probability density functions

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

Justin Bois

Lecturer at the California Institute of Technology

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STATISTICAL THINKING IN PYTHON (PART 1)

Continuous variables

Quantities that can take any value, not just discrete values

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STATISTICAL THINKING IN PYTHON (PART 1)

Michelson's speed of light experiment

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STATISTICAL THINKING IN PYTHON (PART 1)

Michelson's speed of light experiment

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STATISTICAL THINKING IN PYTHON (PART 1)

Probability density function (PDF)

Continuous analog to the PMF Mathematical description of the relative likelihood of

  • bserving a value of a continuous variable
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STATISTICAL THINKING IN PYTHON (PART 1)

Normal PDF

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal PDF

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal CDF

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal CDF

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Let's practice!

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

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Introduction to the Normal distribution

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

Justin Bois

Lecturer at the California Institute of Technology

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal distribution

Describes a continuous variable whose PDF has a single symmetric peak.

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal distribution

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal distribution

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal distribution

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal distribution

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STATISTICAL THINKING IN PYTHON (PART 1)

Normal distribution

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STATISTICAL THINKING IN PYTHON (PART 1)

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STATISTICAL THINKING IN PYTHON (PART 1)

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STATISTICAL THINKING IN PYTHON (PART 1)

Comparing data to a Normal PDF

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STATISTICAL THINKING IN PYTHON (PART 1)

Checking Normality of Michelson data

import numpy as np mean = np.mean(michelson_speed_of_light) std = np.std(michelson_speed_of_light) samples = np.random.normal(mean, std, size=10000) x, y = ecdf(michelson_speed_of_light) x_theor, y_theor = ecdf(samples)

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STATISTICAL THINKING IN PYTHON (PART 1)

Checking Normality of Michelson data

import matplotlib.pyplot as plt import seaborn as sns sns.set() _ = plt.plot(x_theor, y_theor) _ = plt.plot(x, y, marker='.', linestyle='none') _ = plt.xlabel('speed of light (km/s)') _ = plt.ylabel('CDF') plt.show()

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STATISTICAL THINKING IN PYTHON (PART 1)

Checking Normality of Michelson data

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Let's practice!

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

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The Normal distribution: Properties and warnings

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

Justin Bois

Lecturer at the California Institute of Technology

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STATISTICAL THINKING IN PYTHON (PART 1)

Image: Deutsche Bundesbank

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STATISTICAL THINKING IN PYTHON (PART 1)

The Gaussian distribution

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STATISTICAL THINKING IN PYTHON (PART 1)

Length of MA large mouth bass

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STATISTICAL THINKING IN PYTHON (PART 1)

Length of MA large mouth bass

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STATISTICAL THINKING IN PYTHON (PART 1)

Length of MA large mouth bass

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STATISTICAL THINKING IN PYTHON (PART 1)

Mass of MA large mouth bass

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STATISTICAL THINKING IN PYTHON (PART 1)

Light tails of the Normal distribution

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STATISTICAL THINKING IN PYTHON (PART 1)

Light tails of the Normal distribution

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Let's practice!

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

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The Exponential distribution

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

Justin Bois

Lecturer at the California Institute of Technology

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STATISTICAL THINKING IN PYTHON (PART 1)

The Exponential distribution

The waiting time between arrivals of a Poisson process is Exponentially distributed

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STATISTICAL THINKING IN PYTHON (PART 1)

The Exponential PDF

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STATISTICAL THINKING IN PYTHON (PART 1)

Possible Poisson process

Nuclear incidents: Timing of one is independent of all others

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STATISTICAL THINKING IN PYTHON (PART 1)

Exponential inter-incident times

mean = np.mean(inter_times) samples = np.random.exponential(mean, size=10000) x, y = ecdf(inter_times) x_theor, y_theor = ecdf(samples) _ = plt.plot(x_theor, y_theor) _ = plt.plot(x, y, marker='.', linestyle='none') _ = plt.xlabel('time (days)') _ = plt.ylabel('CDF') plt.show()

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STATISTICAL THINKING IN PYTHON (PART 1)

Exponential inter-incident times

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Let's practice!

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

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Final thoughts

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )

Justin Bois

Lecturer at the California Institute of Technology

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STATISTICAL THINKING IN PYTHON (PART 1)

You now can…

Construct (beautiful) instructive plots Compute informative summary statistics Use hacker statistics Think probabilistically

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STATISTICAL THINKING IN PYTHON (PART 1)

In the sequel, you will…

Estimate parameter values Perform linear regressions Compute condence intervals Perform hypothesis tests

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Let's practice!

STATISTIC AL TH IN K IN G IN P YTH ON (PAR T 1 )