🎲Intro to 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 specific values
• The probability of a continuous random variable taking on a specific value is always 0
• Continuous random variables are often used to model real-world phenomena (temperatures, heights, weights)
  • For instance, the weight of a randomly selected apple from a harvest can be modeled as a continuous random variable
• The probability of a continuous random variable falling within a range of values is determined by the area under the curve of its probability density function (PDF)
• Examples of continuous random variables include time, distance, and volume
• The domain of a continuous random variable is an interval of real numbers, which can be bounded or unbounded

Probability Density Functions (PDFs)

• A probability density function (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 curve of a PDF between two points $a$ and $b$ represents the probability of the random variable falling within that range
  • Mathematically, this is expressed as $P(a \leq X \leq b) = \int_a^b f(x) dx$
• The total area under the curve of a PDF is always equal to 1
• PDFs are non-negative functions, meaning $f(x) \geq 0$ for all values of $x$
• The height of a PDF at a specific point does not represent the probability of the random variable taking on that value
  • Instead, the height represents the relative likelihood of the random variable being close to that value
• Examples of PDFs include the normal distribution, exponential distribution, and uniform distribution

Cumulative Distribution Functions (CDFs)

• A cumulative distribution function (CDF) is a function that describes the probability of a continuous random variable being less than or equal to a specific value
• The CDF is denoted as $F(x)$, where $x$ is the value of the continuous random variable
• $F(x) = P(X \leq x) = \int_{-\infty}^x f(t) dt$, where $f(t)$ is the PDF of the random variable
• CDFs are non-decreasing functions, meaning $F(a) \leq F(b)$ if $a \leq b$
• The CDF ranges from 0 to 1, with $F(-\infty) = 0$ and $F(\infty) = 1$
• The probability of a continuous random variable falling within a range $[a, b]$ can be calculated using the CDF
  • $P(a \leq X \leq b) = F(b) - F(a)$
• The PDF can be obtained by differentiating the CDF, $f(x) = \frac{d}{dx}F(x)$

Expected Value and Variance

• The expected value (or mean) of a continuous random variable is a measure of its central tendency
• For a continuous random variable $X$ with PDF $f(x)$, the expected value is given by $E(X) = \int_{-\infty}^{\infty} x f(x) dx$
• The variance of a continuous random variable measures the spread of its distribution around the mean
• The variance is denoted as $Var(X)$ or $\sigma^2$ and is given by $Var(X) = E((X - \mu)^2) = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx$, where $\mu = E(X)$
• The standard deviation, denoted as $\sigma$, is the square root of the variance and has the same units as the random variable
• Properties of expected value and variance for continuous random variables are similar to those for discrete random variables
  • Linearity of expectation: $E(aX + b) = aE(X) + b$, where $a$ and $b$ are constants
  • Variance of a linear transformation: $Var(aX + b) = a^2Var(X)$

Common Continuous Distributions

• Normal (Gaussian) distribution: characterized by its bell-shaped curve and is symmetric about its mean
  • Denoted as $X \sim N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance
  • PDF: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
• Exponential distribution: models the time between events in a Poisson process
  • Denoted as $X \sim Exp(\lambda)$, where $\lambda$ is the rate parameter
  • PDF: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
• Uniform distribution: all values within a specified range are equally likely
  • Denoted as $X \sim U(a, b)$, where $a$ and $b$ are the lower and upper bounds
  • PDF: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
• Other common continuous distributions include the gamma, beta, and Weibull distributions

Applications in Real-World Scenarios

• Continuous random variables are used to model various real-world phenomena (stock prices, waiting times, product lifetimes)
• In finance, stock prices can be modeled using a log-normal distribution, which is based on the normal distribution
• The exponential distribution is often used to model the time between arrivals in a queue or the lifetime of electronic components
• The uniform distribution can be used to model the probability of a dart landing at a specific point on a dartboard
• In quality control, the dimensions of manufactured parts can be modeled using a normal distribution
  • This helps determine the likelihood of a part falling within acceptable tolerance limits
• Continuous random variables are also used in reliability analysis to model the time until failure of a system or component

Solving Problems with Continuous Random Variables

• To solve problems involving continuous random variables, first identify the appropriate distribution and its parameters
• Use the PDF or CDF to calculate probabilities of the random variable falling within specific ranges
• Apply the formulas for expected value and variance to determine the mean and spread of the distribution
• For the normal distribution, use the standard normal (Z) table or calculator to find probabilities
  • Convert the random variable to a standard normal variable using $Z = \frac{X - \mu}{\sigma}$
• When working with linear transformations of continuous random variables, use the properties of expected value and variance
• In some cases, you may need to integrate the PDF to find probabilities or expected values
  • Techniques such as u-substitution, integration by parts, or trigonometric substitution may be required

Key Takeaways and Study Tips

• Understand the difference between discrete and continuous random variables
• Know the properties and interpretations of PDFs and CDFs
• Be able to calculate probabilities using PDFs and CDFs
• Memorize the formulas for expected value and variance of continuous random variables
• Familiarize yourself with the common continuous distributions and their applications
• Practice solving problems using the standard normal table or calculator
• Understand how to apply continuous random variables in real-world scenarios
• Review the properties of expected value and variance for linear transformations
• Practice integrating PDFs to find probabilities and expected values