Course Summary: Introduction to Statistics in Python

Chapter 3 normal distributioin, poisson distribution & expontial distribution

I. Normal Distribution

Concrete Example:

• Example Scenario: The heights of adults in a population typically follow a normal distribution, with most people having heights around the mean and fewer people being much shorter or taller than average.

Python Function: scipy.stats.norm

1. norm.cdf()

   • Definition: The cumulative distribution function (CDF) calculates the probability that a random variable from a normal distribution is less than or equal to a certain value.
   • Example:

     from scipy.stats import norm
     # Probability that a value is less than or equal to 1.96 in a standard normal distribution
     prob = norm.cdf(1.96)
     print(prob)  # Output: 0.9750021048517795

2. norm.pdf()

   • Definition: The probability density function (PDF) calculates the probability density of a normal distribution at a given value.
   • Example:

     from scipy.stats import norm
     # Probability density at value 0 in a standard normal distribution
     density = norm.pdf(0)
     print(density)  # Output: 0.3989422804014327

3. norm.rvs()

   • Definition: The random variate sampling (RVS) function generates random numbers from a normal distribution.
   • Example:

     from scipy.stats import norm
     # Generate 5 random values from a standard normal distribution
     random_values = norm.rvs(size=5)
     print(random_values)

II. Poisson Distribution

Concrete Example:

• Example Scenario: The number of emails a person receives per hour can be modeled using a Poisson distribution if the emails are received independently and at a constant average rate.

Python Function: scipy.stats.poisson

1. poisson.cdf()

   • Definition: The cumulative distribution function (CDF) calculates the probability that the number of events is less than or equal to a certain value.
   • Example:

     from scipy.stats import poisson
     # Probability of receiving 3 or fewer emails in an hour, given the average rate is 2 emails per hour
     prob = poisson.cdf(3, 2)
     print(prob)  # Output: 0.857123460498547

2. poisson.pmf()

   • Definition: The probability mass function (PMF) calculates the probability of a given number of events occurring in a fixed interval.
   • Example:

     from scipy.stats import poisson
     # Probability of receiving exactly 2 emails in an hour, given the average rate is 2 emails per hour
     prob = poisson.pmf(2, 2)
     print(prob)  # Output: 0.2706705664732254

3. poisson.rvs()

   • Definition: The random variate sampling (RVS) function generates random numbers from a Poisson distribution.
   • Example:

     from scipy.stats import poisson
     # Generate 5 random values from a Poisson distribution with an average rate of 2 emails per hour
     random_values = poisson.rvs(2, size=5)
     print(random_values)

III. Exponential Distribution

Concrete Example:

• Example Scenario: The time between arrivals of buses at a bus stop can be modeled using an exponential distribution if the buses arrive independently and at a constant average rate.

Python Function: scipy.stats.expon

1. expon.cdf()

   • Definition: The cumulative distribution function (CDF) calculates the probability that the time between events is less than or equal to a certain value.
   • Example:

     from scipy.stats import expon
     # Probability that the waiting time is less than or equal to 5 minutes, given the average rate is 1 bus per 10 minutes
     prob = expon.cdf(5, scale=10)
     print(prob)  # Output: 0.3934693402873666

2. expon.pdf()

   • Definition: The probability density function (PDF) calculates the probability density of the time between events at a given value.
   • Example:

     from scipy.stats import expon
     # Probability density at 5 minutes, given the average rate is 1 bus per 10 minutes
     density = expon.pdf(5, scale=10)
     print(density)  # Output: 0.09048374180359596

3. expon.rvs()

   • Definition: The random variate sampling (RVS) function generates random numbers from an exponential distribution.
   • Example:

     from scipy.stats import expon
     # Generate 5 random waiting times from an exponential distribution with an average rate of 1 bus per 10 minutes
     random_values = expon.rvs(scale=10, size=5)
     print(random_values)

These summaries provide concrete examples and Python applications of the normal, Poisson, and exponential distributions using the scipy.stats library.