🎲Intro to Probability Unit 9 – Continuous Distributions: Uniform to Normal

Key Concepts and Definitions

• Continuous random variables can take on any value within a specified range or interval
• Probability density functions (PDFs) describe the likelihood of a continuous random variable taking on a particular value
  • PDFs are represented by a curve, where the area under the curve between two points represents the probability of the variable falling within that range
• Cumulative distribution functions (CDFs) give the probability that a continuous random variable is less than or equal to a specific value
• Expected value represents the average value of a continuous random variable over its entire range
• Variance and standard deviation measure the spread or dispersion of a continuous random variable around its expected value

Types of Continuous Distributions

• Uniform distribution characterized by equal probability across a fixed range
• Normal distribution follows a bell-shaped curve, with mean and standard deviation as parameters
• Exponential distribution models the time between events in a Poisson process
• Gamma distribution generalizes the exponential distribution, with shape and scale parameters
• Beta distribution models probabilities over a fixed range, with two shape parameters
• Weibull distribution used in reliability analysis and failure time modeling

The Uniform Distribution

• Uniform distribution denoted as $U(a, b)$, where $a$ and $b$ are the minimum and maximum values of the range
• PDF of a uniform distribution is constant between $a$ and $b$, and zero elsewhere: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
• CDF of a uniform distribution is a linear function between $a$ and $b$: $F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$
• Expected value of a uniform distribution is the midpoint of the range: $E(X) = \frac{a+b}{2}$
• Variance of a uniform distribution is $Var(X) = \frac{(b-a)^2}{12}$

The Normal Distribution

• Normal distribution denoted as $N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance
• PDF of a normal distribution is a bell-shaped curve: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
• CDF of a normal distribution has no closed-form expression and is typically calculated using tables or software
• Standard normal distribution has a mean of 0 and a standard deviation of 1, denoted as $Z \sim N(0, 1)$
  • Any normal distribution can be standardized using the z-score: $Z = \frac{X-\mu}{\sigma}$
• 68-95-99.7 rule approximates the percentage of data within 1, 2, and 3 standard deviations of the mean in a normal distribution

Properties and Characteristics

• Continuous distributions have an infinite number of possible values within a range
• The area under the PDF curve between two points represents the probability of the variable falling within that range
• The total area under the PDF curve is always equal to 1
• The mean, median, and mode of a normal distribution are all equal
• Continuous distributions can be transformed using functions, resulting in new distributions with different properties

Probability Calculations

• Probability calculations for continuous distributions involve integrating the PDF over a specified range
  • $P(a \leq X \leq b) = \int_a^b f(x) dx$
• For uniform distributions, probabilities can be calculated using the CDF: $P(X \leq x) = F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$
• For normal distributions, probabilities are often calculated using z-scores and standard normal tables
  • First, standardize the variable: $Z = \frac{X-\mu}{\sigma}$
  • Then, use tables or software to find the probability: $P(X \leq x) = P(Z \leq \frac{x-\mu}{\sigma})$

Real-World Applications

• Uniform distributions model random number generation and sampling in computer science
• Normal distributions appear in natural phenomena (heights, weights) and measurement errors
• Financial returns often follow a log-normal distribution, with stock prices modeled using geometric Brownian motion
• Exponential distributions model waiting times between events (customer arrivals, radioactive decay)
• Gamma distributions used in queuing theory and reliability analysis
• Beta distributions model proportions and probabilities (election outcomes, market share)

Practice Problems and Examples

• Calculate the probability that a uniform random variable $X \sim U(2, 7)$ is between 3 and 5
  • $P(3 \leq X \leq 5) = \frac{5-3}{7-2} = \frac{2}{5}$
• Find the 90th percentile of a normal distribution with mean 50 and standard deviation 10
  • 90th percentile corresponds to $z = 1.28$, so $x = \mu + z\sigma = 50 + 1.28(10) = 62.8$
• Determine the expected value and variance of a uniform distribution $U(-3, 6)$
  • $E(X) = \frac{-3+6}{2} = 1.5$ and $Var(X) = \frac{(6-(-3))^2}{12} = \frac{81}{12} = 6.75$
• Calculate the probability that a standard normal random variable is greater than 1.5
  • $P(Z > 1.5) = 1 - P(Z \leq 1.5) \approx 1 - 0.9332 = 0.0668$