🎲Data Science Statistics Unit 5 – Continuous Probability Distributions

Key Concepts and Definitions

• Continuous probability distributions describe the probabilities of outcomes for a continuous random variable
• Random variables that can take on any value within a specified range are considered continuous
• Probability density functions (PDFs) used to specify the probability of a continuous random variable falling within a particular range of values
• Cumulative distribution functions (CDFs) calculate the probability that a continuous random variable takes a value less than or equal to a given value
• Expected value represents the average value of a continuous random variable over an infinite number of trials
• Variance measures the average squared deviation of a continuous random variable from its expected value
• Skewness quantifies the asymmetry of a continuous probability distribution
  • Positive skewness indicates a longer tail on the right side of the distribution
  • Negative skewness indicates a longer tail on the left side of the distribution
• Kurtosis measures the heaviness of the tails and peakedness of a continuous probability distribution compared to a normal distribution

Types of Continuous Distributions

• Normal distribution (Gaussian distribution) characterized by a symmetric bell-shaped curve with mean $\mu$ and standard deviation $\sigma$
• Exponential distribution models the time between events in a Poisson process, with parameter $\lambda$ representing the rate of occurrence
• Gamma distribution represents the waiting time until a specified number of events occur in a Poisson process
  • Shape parameter $k$ and scale parameter $\theta$ determine the distribution's form
• Beta distribution models probabilities or proportions bounded between 0 and 1, with shape parameters $\alpha$ and $\beta$
• Uniform distribution assigns equal probability to all values within a specified range $[a, b]$
• Weibull distribution used to model failure rates and reliability, with shape parameter $k$ and scale parameter $\lambda$
• Lognormal distribution describes variables whose logarithm follows a normal distribution, useful for modeling financial data and particle sizes
• Cauchy distribution has a symmetric bell shape with heavy tails, lacking finite moments such as mean and variance

Properties of Continuous Distributions

• Probabilities for continuous distributions are represented by areas under the curve of the probability density function (PDF)
• The total area under the PDF curve is always equal to 1, representing the sum of all probabilities
• Continuous distributions have an infinite number of possible outcomes within a given range
• The probability of a continuous random variable taking on any specific value is 0, as there are infinitely many possible values
• Continuous distributions can be transformed using functions such as logarithms or powers to create new distributions
• The central limit theorem states that the sum of a large number of independent and identically distributed random variables will approximate a normal distribution
• Moment generating functions (MGFs) used to calculate moments of a continuous distribution, such as mean and variance
• Continuous distributions can be truncated or censored to focus on a specific range of values or to account for missing data

Probability Density Functions (PDFs)

• Probability density functions (PDFs) specify the relative likelihood of a continuous random variable taking on a specific value
• PDFs are non-negative functions that integrate to 1 over the entire domain of the random variable
• The probability of a continuous random variable falling within a given range is determined by integrating the PDF over that range
• PDFs can be used to calculate probabilities, quantiles, and other properties of continuous distributions
• The mode of a continuous distribution is the value at which the PDF reaches its maximum
• PDFs are often expressed using mathematical formulas specific to each type of continuous distribution
  • Normal distribution PDF: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
  • Exponential distribution PDF: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
• Kernel density estimation (KDE) is a non-parametric method for estimating the PDF of a continuous random variable based on a finite sample of data points

Cumulative Distribution Functions (CDFs)

• Cumulative distribution functions (CDFs) calculate the probability that a continuous random variable takes a value less than or equal to a given value
• CDFs are monotonically increasing functions that map the domain of the random variable to the interval $[0, 1]$
• The CDF is the integral of the probability density function (PDF) from negative infinity to a specific value
• CDFs can be used to calculate probabilities, quantiles, and other properties of continuous distributions
• The median of a continuous distribution is the value at which the CDF equals 0.5
• Inverse CDFs (quantile functions) used to determine the value of a random variable corresponding to a given probability
• CDFs are often expressed using mathematical formulas specific to each type of continuous distribution
  • Normal distribution CDF: $F(x) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]$
  • Exponential distribution CDF: $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$

Expected Values and Moments

• The expected value (mean) of a continuous random variable is the average value obtained over an infinite number of trials
  • Calculated by integrating the product of the random variable and its probability density function (PDF) over the entire domain
• Variance measures the average squared deviation of a continuous random variable from its expected value
  • Calculated by integrating the product of the squared deviation and the PDF over the entire domain
• Standard deviation is the square root of the variance, providing a measure of dispersion in the same units as the random variable
• Skewness is the third standardized moment, quantifying the asymmetry of a continuous probability distribution
  • Positive skewness indicates a longer tail on the right side, while negative skewness indicates a longer tail on the left side
• Kurtosis is the fourth standardized moment, measuring the heaviness of the tails and peakedness of a continuous probability distribution compared to a normal distribution
  • Higher kurtosis indicates heavier tails and a sharper peak, while lower kurtosis indicates lighter tails and a flatter peak
• Moment generating functions (MGFs) are used to calculate moments of a continuous distribution by taking derivatives of the MGF evaluated at 0

Applications in Data Science

• Continuous probability distributions used to model and analyze real-world phenomena in various domains, such as finance, physics, and engineering
• Normal distribution commonly used to model errors, noise, and natural variations in data
  • Central limit theorem justifies the use of normal distribution for modeling the sum or average of a large number of independent variables
• Exponential and gamma distributions used to model waiting times, such as the time between customer arrivals or the duration of a phone call
• Beta distribution used to model probabilities or proportions, such as the click-through rate of an online advertisement or the success rate of a medical treatment
• Lognormal distribution used to model financial variables (stock prices) and physical quantities (particle sizes) that exhibit a skewed distribution
• Kernel density estimation (KDE) used to estimate the underlying probability density function of a dataset, helping to identify patterns and anomalies
• Continuous distributions used in hypothesis testing and confidence interval estimation for population parameters
• Bayesian inference relies on continuous distributions to represent prior beliefs and update them based on observed data

Practical Examples and Exercises

• Analyzing the distribution of heights in a population using a normal distribution
  • Calculating the probability of an individual being taller than a specific height
  • Determining the height corresponding to a given percentile
• Modeling the time between customer arrivals at a store using an exponential distribution
  • Estimating the probability of no customers arriving within a specific time frame
  • Calculating the expected number of customers arriving within a given period
• Assessing the reliability of a product using a Weibull distribution
  • Determining the probability of a product failing before a specified time
  • Estimating the mean time to failure for the product
• Analyzing the distribution of test scores using a beta distribution
  • Calculating the probability of a student scoring within a specific range
  • Determining the minimum score required to be in the top 10% of the class
• Modeling the distribution of incomes in a population using a lognormal distribution
  • Estimating the median income and the income inequality (Gini coefficient)
  • Calculating the probability of an individual earning more than a specific amount
• Applying kernel density estimation (KDE) to visualize the underlying distribution of a dataset
  • Identifying potential subgroups or clusters within the data
  • Detecting outliers or anomalies that deviate significantly from the estimated distribution