📉Intro to Business Statistics Unit 5 – Continuous Random Variables

What Are Continuous Random Variables?

• Continuous random variables can take on any value within a specified range or interval
• Unlike discrete random variables, continuous random variables are not limited to whole numbers or integers
• The possible values of a continuous random variable form a continuous range, such as all real numbers between 0 and 1
• Continuous random variables are often used to model measurements, such as height, weight, time, or temperature
• The probability of a continuous random variable taking on a specific value is always 0, as there are infinitely many possible values within any given range

Key Concepts and Definitions

• Sample space: The set of all possible outcomes of a random experiment
• Event: A subset of the sample space, representing a specific outcome or group of outcomes
• Probability: A measure of the likelihood that an event will occur, expressed as a value between 0 and 1
• Probability density function (PDF): A function that describes the relative likelihood of a continuous random variable taking on a specific value
  • The area under the PDF curve between two points represents the probability of the variable falling within that range
• Cumulative distribution function (CDF): A function that describes the probability that a continuous random variable will take on a value less than or equal to a given value
  • The CDF is the integral of the PDF from negative infinity to the given value

Probability Density Functions (PDFs)

• A PDF is a function that describes the relative likelihood of a continuous random variable taking on a specific value
• The PDF is denoted as $f(x)$, where $x$ is the value of the continuous random variable
• The area under the PDF curve between two points, $a$ and $b$, represents the probability of the variable falling within that range, denoted as $P(a \leq X \leq b)$
• Properties of a PDF:
  • The PDF is always non-negative: $f(x) \geq 0$ for all $x$
  • The total area under the PDF curve is equal to 1: $\int_{-\infty}^{\infty} f(x) dx = 1$
• To find the probability of a continuous random variable falling within a specific range, integrate the PDF over that range: $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$

Cumulative Distribution Functions (CDFs)

• A CDF is a function that describes the probability that a continuous random variable will take on a value less than or equal to a given value
• The CDF is denoted as $F(x)$, where $x$ is the value of the continuous random variable
• The CDF is the integral of the PDF from negative infinity to the given value: $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$
• Properties of a CDF:
  • The CDF is always non-decreasing: If $a \leq b$, then $F(a) \leq F(b)$
  • The CDF approaches 0 as $x$ approaches negative infinity: $\lim_{x \to -\infty} F(x) = 0$
  • The CDF approaches 1 as $x$ approaches positive infinity: $\lim_{x \to \infty} F(x) = 1$
• To find the probability of a continuous random variable falling within a specific range using the CDF, subtract the CDF values at the endpoints: $P(a \leq X \leq b) = F(b) - F(a)$

Expected Value and Variance

• The expected value (or mean) of a continuous random variable is a measure of its central tendency, denoted as $E(X)$ or $\mu$
  • It is calculated by integrating the product of the variable and its PDF over the entire range: $E(X) = \int_{-\infty}^{\infty} x f(x) dx$
• The variance of a continuous random variable is a measure of its dispersion, denoted as $Var(X)$ or $\sigma^2$
  • It is calculated by integrating the product of the squared deviation from the mean and the PDF over the entire range: $Var(X) = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx$
• The standard deviation is the square root of the variance, denoted as $\sigma$
• The expected value and variance provide insights into the typical behavior and spread of a continuous random variable

Common Continuous Distributions

• Normal (Gaussian) distribution: Symmetric bell-shaped curve characterized by its mean $\mu$ and standard deviation $\sigma$
  • PDF: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
  • CDF: No closed-form expression; typically approximated using tables or software
• Exponential distribution: Models the time between events in a Poisson process, characterized by its rate parameter $\lambda$
  • PDF: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
  • CDF: $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$
• Uniform distribution: Constant probability over a specified interval $[a, b]$
  • PDF: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
  • CDF: $F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$

Applications in Business

• Modeling customer arrival times or service times in a queueing system using the exponential distribution
• Analyzing the distribution of product defects or failures using the normal distribution
• Estimating the time to complete a project or the duration of a task using the uniform distribution
• Assessing financial risk by modeling asset returns or portfolio values using continuous distributions
• Optimizing inventory levels by considering the distribution of demand or lead times

Problem-Solving Techniques

• Identify the type of continuous random variable and its parameters based on the given information
• Determine the appropriate distribution (normal, exponential, uniform, etc.) to model the problem
• Use the PDF or CDF to calculate probabilities, either by integrating the PDF or using the CDF formula
• Apply the expected value and variance formulas to determine the mean and dispersion of the continuous random variable
• Interpret the results in the context of the business problem, making decisions or drawing conclusions based on the calculated probabilities or statistics