Intro to Probability

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

Continuous distributions are a crucial concept in probability theory, describing random variables that can take on any value within a range. This unit focuses on two key distributions: uniform and normal, exploring their properties, probability density functions, and applications. Understanding continuous distributions is essential for modeling real-world phenomena. The uniform distribution represents equal likelihood across a range, while the normal distribution's bell-shaped curve is ubiquitous in nature and statistics, forming the foundation for many statistical analyses and hypothesis tests.

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)U(a, b), where aa and bb are the minimum and maximum values of the range
  • PDF of a uniform distribution is constant between aa and bb, and zero elsewhere: f(x)=1baf(x) = \frac{1}{b-a} for axba \leq x \leq b
  • CDF of a uniform distribution is a linear function between aa and bb: F(x)=xabaF(x) = \frac{x-a}{b-a} for axba \leq x \leq b
  • Expected value of a uniform distribution is the midpoint of the range: E(X)=a+b2E(X) = \frac{a+b}{2}
  • Variance of a uniform distribution is Var(X)=(ba)212Var(X) = \frac{(b-a)^2}{12}

The Normal Distribution

  • Normal distribution denoted as N(μ,σ2)N(\mu, \sigma^2), where μ\mu is the mean and σ2\sigma^2 is the variance
  • PDF of a normal distribution is a bell-shaped curve: f(x)=1σ2πe(xμ)22σ2f(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 ZN(0,1)Z \sim N(0, 1)
    • Any normal distribution can be standardized using the z-score: Z=XμσZ = \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(aXb)=abf(x)dxP(a \leq X \leq b) = \int_a^b f(x) dx
  • For uniform distributions, probabilities can be calculated using the CDF: P(Xx)=F(x)=xabaP(X \leq x) = F(x) = \frac{x-a}{b-a} for axba \leq x \leq b
  • For normal distributions, probabilities are often calculated using z-scores and standard normal tables
    • First, standardize the variable: Z=XμσZ = \frac{X-\mu}{\sigma}
    • Then, use tables or software to find the probability: P(Xx)=P(Zxμσ)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 XU(2,7)X \sim U(2, 7) is between 3 and 5
    • P(3X5)=5372=25P(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=1.28z = 1.28, so x=μ+zσ=50+1.28(10)=62.8x = \mu + z\sigma = 50 + 1.28(10) = 62.8
  • Determine the expected value and variance of a uniform distribution U(3,6)U(-3, 6)
    • E(X)=3+62=1.5E(X) = \frac{-3+6}{2} = 1.5 and Var(X)=(6(3))212=8112=6.75Var(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)=1P(Z1.5)10.9332=0.0668P(Z > 1.5) = 1 - P(Z \leq 1.5) \approx 1 - 0.9332 = 0.0668


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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