Normal distributions F OUN DATION S OF P ROBABILITY IN P YTH ON - - PowerPoint PPT Presentation

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Normal distributions F OUN DATION S OF P ROBABILITY IN P YTH ON - - PowerPoint PPT Presentation

Jul 11, 2023 94 likes •664 views

Normal distributions F OUN DATION S OF P ROBABILITY IN P YTH ON Alexander A. Ramrez M. CEO @ Synergy Vision Modeling for measures FOUNDATIONS OF PROBABILITY IN PYTHON Adults' heights example FOUNDATIONS OF PROBABILITY IN PYTHON

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Normal distributions

F OUN DATION S OF P ROBABILITY IN P YTH ON

Alexander A. Ramírez M.

CEO @ Synergy Vision

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FOUNDATIONS OF PROBABILITY IN PYTHON

Modeling for measures

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Adults' heights example

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Probability density

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Probability density examples

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Probability density and probability

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Symmetry

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Mean

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Mean (Cont.)

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Mean (Cont.)

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Standard deviation

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Standard deviation (Cont.)

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Standard deviation (Cont.)

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One standard deviation

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Two standard deviations

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Three standard deviations

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Normal sampling

# Import norm, matplotlib.pyplot, and seaborn from scipy.stats import norm import matplotlib.pyplot as plt import seaborn as sns # Create the sample using norm.rvs() sample = norm.rvs(loc=0, scale=1, size=10000, random_state=13) # Plot the sample sns.distplot(sample) plt.show()

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Normal sampling (Cont.)

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Let's do some exercises with normal distributions

F OUN DATION S OF P ROBABILITY IN P YTH ON

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Normal probabilities

F OUN DATION S OF P ROBABILITY IN P YTH ON

Alexander A. Ramírez M.

CEO @ Synergy Vision

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FOUNDATIONS OF PROBABILITY IN PYTHON

Probability density

In Python this can be done in a couple of lines:

# Import norm from scipy.stats import norm # Calculate the probability density # with pdf norm.pdf(-1, loc=0, scale=1) 0.24197072451914337

loc parameter species the mean and scale

parameter species the standard deviation.

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pdf() vs. cdf()

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pdf() vs. cdf() (Cont.)

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pdf() vs. cdf() (Cont.)

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FOUNDATIONS OF PROBABILITY IN PYTHON

Cumulative distribution function examples

# Calculate cdf of -1 norm.cdf(-1) 0.15865525393145707 # Calculate cdf of 0.5 norm.cdf(0.5) 0.6914624612740131

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The percent point function (ppf)

# Calculate ppf of 0.2 norm.ppf(0.2)

  • 0.8416212335729142

# Calculate ppf of 55% norm.ppf(0.55) 0.12566134685507416

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ppf() is the inverse of cdf()

# Calculate cdf of value 0 norm.cdf(0) 0.5 # Calculate ppf of probability 50% norm.ppf(0.5)

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Probability between two values

# Create our variables a = -1 b = 1 # Calculate the probability between # two values, subtracting norm.cdf(b) - norm.cdf(a) 0.6826894921370859

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Tail probability

# Create our variable a = 1 # Calculate the complement # of cdf() using sf() norm.sf(a) 0.15865525393145707

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Tails

# Create our variables a = -2 b = 2 # Calculate tail probability # by adding each tail norm.cdf(a) + norm.sf(b) 0.04550026389635839

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Tails (Cont.)

# Create our variables a = -2 b = 2 # Calculate tail probability # by adding each tail norm.cdf(a) + norm.sf(b) 0.04550026389635839

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Intervals

# Create our variable alpha = 0.95 # Calculate the interval norm.interval(alpha) (-1.959963984540054, 1.959963984540054)

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On to some practice!

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Poisson distributions

F OUN DATION S OF P ROBABILITY IN P YTH ON

Alexander A. Ramírez M.

CEO @ Synergy Vision

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Poisson modeling

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Poisson distribution properties

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Probability mass function (pmf)

Imagine you have 2.2 calls per minute.

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Probability mass function (pmf) (Cont.)

In Python we do the following:

# Import poisson from scipy.stats import poisson # Calculate the probability mass # with pmf poisson.pmf(k=3, mu=2.2) 0.19663867170702193

mu parameter species the mean of successful

events

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pmf examples

# Calculate pmf of 0 poisson.pmf(k=0, mu=2.2) 0.11080315836233387 # Calculate pmf of 6 poisson.pmf(k=6, mu=2.2) 0.01744840480280308

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Different means

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Cumulative distribution function (cdf)

# Calculate cdf of 2 poisson.cdf(k=2, mu=2.2) 0.6227137499963162 # Calculate cdf of 5 poisson.cdf(k=5, mu=2.2) 0.9750902496952996

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Survival function and percent point function (ppf)

# Calculate sf of 2 poisson.sf(k=2, mu=2.2) 0.3772862500036838 # Calculate ppf of 0.5 poisson.ppf(q=0.5, mu=2.2) 2.0

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Sample generation (rvs)

# Import poisson, matplotlib.pyplot, and seaborn from scipy.stats import poisson import matplotlib.pyplot as plt import seaborn as sns # Create the sample using poisson.rvs() sample = poisson.rvs(mu=2.2, size=10000, random_state=13) # Plot the sample sns.distplot(sample, kde=False) plt.show()

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Sample generation (Cont.)

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Let's practice with Poisson

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Geometric distributions

F OUN DATION S OF P ROBABILITY IN P YTH ON

Alexander A. Ramírez M.

CEO @ Synergy Vision

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FOUNDATIONS OF PROBABILITY IN PYTHON

Geometric modeling

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Geometric parameter

Model for a basketball player with probability 0.3

  • f scoring.

We can model a grizzly bear that has a 0.033 probability of catching a salmon.

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Probability mass function (pmf)

In Python we code this as follows:

# Import geom from scipy.stats import geom # Calculate the probability mass # with pmf geom.pmf(k=30, p=0.0333) 0.02455102908739612

p parameter species probability of success.

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Cumulative distribution function (cdf)

# Calculate cdf of 4 geom.cdf(k=4, p=0.3) 0.7598999999999999

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Survival function (sf)

# Calculate sf of 2 geom.sf(k=2, p=0.3) 0.49000000000000005

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Percent point function (ppf)

# Calculate ppf of 0.6 geom.ppf(q=0.6, p=0.3) 3.0

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Sample generation (rvs)

# Import poisson, matplotlib.pyplot, and seaborn from scipy.stats import geom import matplotlib.pyplot as plt import seaborn as sns # Create the sample using geom.rvs() sample = geom.rvs(p=0.3, size=10000, random_state=13) # Plot the sample sns.distplot(sample, bins = np.linspace(0,20,21), kde=False) plt.show()

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Sample generation (rvs) (Cont.)

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Let's go try until we succeed!

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