9.8 Probability Distribution Function of Continuous Distributions

A Probability Distribution Function (PDF) for a continuous random variable describes the likelihood of that variable taking on a specific value within a given range. Unlike discrete distributions, continuous distributions can assume an infinite number of values within an interval. The PDF quantifies the probability density across this range.

I. Continuous Uniform Distribution

The Continuous Uniform Distribution models scenarios where all outcomes within a specified interval are equally likely. Its PDF is represented by a constant value across this interval.

PDF:

f(x) = 1 / (b - a), for a ≤ x ≤ b

Where:

• a: The lower bound of the interval.
• b: The upper bound of the interval.
• x: A value within the interval [a, b].

Description: This distribution is employed when there's no bias towards any particular outcome within the defined range [a, b].

Example: Imagine randomly selecting a number between 0 and 10. Each number in this range has an equal chance of being chosen.

II. Gamma Distribution

The Gamma Distribution is used to model the waiting time until a specified number of events occur in a Poisson process. It is characterized by two key parameters: a shape parameter ($\alpha$) and a rate parameter ($\lambda$).

PDF:

f(x) = (λ^α / Γ(α)) * x^(α - 1) * e^(-λx), for x > 0

Where:

• α ($\alpha$): The shape parameter.
• λ ($\lambda$): The rate parameter.
• Γ(α) ($\Gamma(\alpha)$): The Gamma function, a generalization of the factorial function.
• x: The value of the random variable, where x > 0.

Description: This distribution is frequently applied in reliability analysis and queuing theory to model durations or waiting times.

III. Exponential Distribution

The Exponential Distribution is a special case of the Gamma Distribution (when $\alpha = 1$). It models the time elapsed between successive events in a Poisson process, assuming a constant average rate of events.

PDF:

f(x; λ) = λ * e^(-λx), for x ≥ 0

Where:

• λ ($\lambda$): The rate parameter, representing the average number of events per unit of time.
• x: The time between events, where x ≥ 0.

Description: Widely used in survival analysis, reliability engineering, and queue modeling to describe the time until an event occurs.

Example: The time between customer arrivals at a service counter, assuming arrivals occur at a constant average rate.

IV. Chi-Square Distribution

The Chi-Square ($\chi^2$) Distribution is a fundamental tool in hypothesis testing, particularly for analyzing variances and tests of independence (such as the chi-square test).

PDF:

f(x) = (1 / (2^(ν/2) * Γ(ν/2))) * x^((ν/2) - 1) * e^(-x/2), for x > 0

Where:

• ν ($\nu$): The degrees of freedom, which determines the shape of the distribution.
• Γ(ν/2) ($\Gamma(\nu/2)$): The Gamma function evaluated at ν/2.
• x: The value of the random variable, where x > 0.

Description: This distribution is typically right-skewed. As the degrees of freedom (ν) increase, the distribution becomes more symmetric and approaches a normal distribution.

Example: Testing if the observed frequencies of categories in a dataset match expected frequencies (e.g., a survey result matching a known population distribution).

V. Beta Distribution

The Beta Distribution is a versatile distribution defined on the interval [0, 1]. It is characterized by two shape parameters, $\alpha$ and $\beta$, allowing for a wide variety of shapes. It is commonly used for modeling proportions and probabilities.

PDF:

f(x) = (Γ(α + β) / (Γ(α) * Γ(β))) * x^(α - 1) * (1 - x)^(β - 1), for 0 < x < 1

Where:

• α ($\alpha$): A shape parameter.
• β ($\beta$): Another shape parameter.
• Γ ($\Gamma$): The Gamma function.
• x: The value of the random variable, where 0 < x < 1.

Description: Its flexibility makes it ideal for applications in Bayesian statistics and for modeling proportions or probabilities that are bounded between 0 and 1.

Example: Modeling the proportion of successful trials in a series of experiments, or the probability of a user clicking on an advertisement.

VI. Lognormal Distribution

The Lognormal Distribution describes a variable whose natural logarithm is normally distributed. This distribution is particularly useful for modeling variables that exhibit multiplicative growth or have a skewed distribution with a long tail towards larger values.

PDF:

f(x) = (1 / (x * σ * √(2π))) * e^(-(ln(x) - μ)^2 / (2σ^2)), for x > 0

Where:

• μ ($\mu$): The mean of the natural logarithm of the variable.
• σ ($\sigma$): The standard deviation of the natural logarithm of the variable.
• x: The value of the random variable, where x > 0.

Description: The Lognormal distribution is strictly positive and right-skewed. It is often used in modeling financial data, biological measurements, and physical phenomena.

Example: Modeling stock prices, income distribution, or the size of particles in a mixture.

VII. Normal Distribution

The Normal Distribution, also known as the Gaussian distribution or bell curve, is arguably the most important continuous probability distribution. Its significance stems from the Central Limit Theorem. It is defined by its mean ($\mu$) and standard deviation ($\sigma$).

PDF:

f(x) = (1 / (σ * √(2π))) * e^(-(x - μ)^2 / (2σ^2)), for -∞ < x < ∞

Where:

• μ ($\mu$): The mean, which represents the center of the distribution.
• σ ($\sigma$): The standard deviation, which measures the spread or dispersion of the data.
• x: The value of the random variable, ranging from negative infinity to positive infinity (x ∈ (−∞, ∞)).

Description: The Normal Distribution is characterized by its symmetric, bell-shaped curve. It is widely applied across natural and social sciences, quality control, and inferential statistics due to its widespread appearance in data and its crucial role in statistical inference.

Example: Heights of adult humans, measurement errors in scientific experiments, and test scores often approximate a normal distribution.

Summary Table

Frequently Asked Questions

• What is a probability distribution function (PDF) for continuous random variables? A PDF for continuous random variables describes the relative likelihood for a random variable to take on a given value. The area under the PDF curve over an interval represents the probability that the variable falls within that interval.

• How is the Continuous Uniform Distribution defined and where is it used? It's defined by f(x) = 1 / (b - a) for a ≤ x ≤ b. It's used when all outcomes within an interval are equally probable, like selecting a random number from a specific range.

• What are the key parameters of the Gamma Distribution and what does it model? The key parameters are the shape parameter ($\alpha$) and the rate parameter ($\lambda$). It models waiting times for a specified number of events in a Poisson process.

• Explain the relationship between the Exponential and Gamma distributions. The Exponential distribution is a special case of the Gamma distribution where the shape parameter $\alpha$ is equal to 1. Both model waiting times between events in a Poisson process.

• Describe the probability density function of the Chi-Square Distribution. Its PDF is f(x) = (1 / (2^(ν/2) * Γ(ν/2))) * x^((ν/2) - 1) * e^(-x/2) for x > 0, where ν is the degrees of freedom. It's crucial for variance analysis and goodness-of-fit tests.

• What is the significance of the Beta Distribution and what interval is it defined on? The Beta distribution is significant for its flexibility in modeling proportions and probabilities, as it can take on many shapes depending on its α and β parameters. It is defined on the interval (0, 1).

• How is the Lognormal Distribution different from the Normal Distribution? In a Lognormal distribution, the logarithm of the variable is normally distributed, leading to a right-skewed distribution for the variable itself. The Normal distribution is symmetric and defined for all real numbers.

• What does the Normal Distribution represent and why is it widely used? The Normal distribution represents data that clusters symmetrically around a mean. It's widely used because many natural phenomena follow this distribution, and the Central Limit Theorem states that the sum (or average) of a large number of independent random variables will tend towards a normal distribution, regardless of their original distributions.

• How does the Central Limit Theorem relate to the Normal Distribution? The Central Limit Theorem is a cornerstone of statistics that explains why the Normal distribution is so prevalent. It states that the sampling distribution of the sample mean (or sum) will approach a normal distribution as the sample size becomes large, irrespective of the population's distribution.

• What is the importance of the gamma function in continuous distributions? The Gamma function ($\Gamma(z)$) is a generalization of the factorial function to complex and real numbers. It appears in the normalization constants of several continuous probability distributions (like Gamma, Beta, and Chi-Square), ensuring that the total probability over the entire domain integrates to 1.