5.4 Continuous Distribution

Continuous distributions model random variables that can take any value within a range. They're crucial for analyzing real-world phenomena like heights, weights, and waiting times. Unlike discrete distributions, probabilities are calculated using integrals rather than summing individual outcomes.

Key concepts include probability density functions (PDFs), cumulative distribution functions (CDFs), and expected values. We'll explore uniform and exponential distributions, interpret density functions, and analyze continuous distributions using means and variances. These tools help us understand and predict continuous random variables.

Continuous Distributions

Continuous probability calculations

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• Continuous uniform distribution
  • Probability density function (PDF) defines the probability of a random variable taking a value within a specific range $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
  • Cumulative distribution function (CDF) gives the probability of a random variable being less than or equal to a specific value $F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$
  • Probability calculation involves finding the area under the PDF curve between two values $P(c \leq X \leq d) = \frac{d-c}{b-a}$ for $a \leq c \leq d \leq b$ (rolling a fair die, selecting a random number between 0 and 1)
• Exponential distribution
  • Probability density function (PDF) models the time between events in a Poisson process $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
  • Cumulative distribution function (CDF) gives the probability of an event occurring before a specific time $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$
  • Probability calculation involves finding the probability of an event occurring after a specific time $P(X > x) = e^{-\lambda x}$ for $x \geq 0$ (time between customer arrivals, lifespan of a light bulb)
  • Memoryless property states that the probability of an event occurring after a specific time is independent of the time already elapsed $P(X > s + t | X > s) = P(X > t)$ for $s, t \geq 0$

Interpretation of density functions

• Probability density function (PDF)
  • Non-negative function $f(x)$ defined over the range of the random variable $X$ (height, weight, temperature)
  • Area under the curve of $f(x)$ over an interval represents the probability of $X$ falling within that interval (probability of a person's height being between 160cm and 180cm)
  • Total area under the curve of $f(x)$ is equal to 1 (sum of all probabilities must be 1)
• Cumulative distribution function (CDF)
  • Function $F(x)$ that gives the probability of the random variable $X$ taking a value less than or equal to $x$ (probability of a person's weight being less than or equal to 70kg)
  • $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$ (integral of the PDF from negative infinity to $x$)
  • Properties: non-decreasing (probability increases as $x$ increases), right-continuous (no jumps in probability), and $\lim_{x \to -\infty} F(x) = 0$ (probability approaches 0 as $x$ approaches negative infinity) and $\lim_{x \to \infty} F(x) = 1$ (probability approaches 1 as $x$ approaches infinity)
  • Quantile function is the inverse of the CDF, used to find specific percentiles of a distribution

Analysis of continuous distributions

• Expected value (mean) of a continuous random variable $X$
  • $E(X) = \int_{-\infty}^{\infty} x f(x) dx$ (weighted average of all possible values of $X$)
  • Linearity of expectation: $E(aX + b) = aE(X) + b$ for constants $a$ and $b$ (scaling and shifting the distribution)
• Variance of a continuous random variable $X$
  • $Var(X) = E[(X - \mu)^2] = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx$, where $\mu = E(X)$ (average squared deviation from the mean)
  • Alternative formula: $Var(X) = E(X^2) - [E(X)]^2$ (expected value of the squared random variable minus the squared expected value)
  • Properties: $Var(aX + b) = a^2 Var(X)$ for constants $a$ and $b$ (scaling the distribution)
• Standard deviation: $\sigma = \sqrt{Var(X)}$ (square root of the variance, measures the spread of the distribution)

Advanced Concepts in Continuous Distributions

• Normal distribution (also known as Gaussian distribution) is a symmetric, bell-shaped distribution characterized by its mean and standard deviation
• Moment-generating function is used to calculate moments of a distribution and determine its probability distribution
• Kurtosis measures the tailedness of a distribution, indicating whether data are heavy-tailed or light-tailed relative to a normal distribution
• Likelihood function is used in parameter estimation, representing how likely a set of parameters is for a given set of observations

Applying Continuous Distributions

Continuous probability calculations

• Examples of continuous uniform distribution applications
  • Modeling the time a bus arrives within a specified interval (arrival time between 7:00am and 7:15am)
  • Analyzing the distribution of a product's dimensions within tolerance limits (length of a bolt between 9.9cm and 10.1cm)
• Examples of exponential distribution applications
  • Modeling the time between customer arrivals in a queue (time between customers entering a store)
  • Analyzing the lifespan of electronic components (time until a computer hard drive fails)
• Solving problems using the given PDF, CDF, and probability formulas (calculating the probability of a bus arriving within 5 minutes of the scheduled time, finding the probability of a customer arriving within 10 minutes)

Interpretation of density functions

• Identifying the key features of a PDF
  • Domain (range of possible values), range (range of possible probabilities), and any discontinuities (gaps in the distribution)
  • Peaks (modes), valleys (antimodes), and symmetry (reflection across the mean)
  • Skewness (asymmetry) and modality (number of peaks) (unimodal, bimodal, etc.)
• Interpreting the meaning of a CDF
  • Understanding the relationship between the CDF and the PDF (CDF is the integral of the PDF)
  • Using the CDF to calculate probabilities (probability of a random variable being less than or equal to a specific value) and percentiles (value below which a certain percentage of observations fall)

Analysis of continuous distributions

• Calculating the mean and variance of continuous distributions
  • Using the formulas for expected value and variance (integrating the PDF multiplied by $x$ or $(x - \mu)^2$)
  • Applying the linearity of expectation and properties of variance (scaling and shifting the distribution)
• Comparing the means and variances of different continuous distributions
  • Understanding the effects of distribution parameters on the mean and variance (changing the rate parameter $\lambda$ in an exponential distribution)
  • Using the mean and variance to make decisions or draw conclusions in real-world scenarios (comparing the average waiting times of two different queuing systems, determining which product has more consistent dimensions)