🎲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.
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)=b−a1 for a≤x≤b
CDF of a uniform distribution is a linear function between a and b: F(x)=b−ax−a for a≤x≤b
Expected value of a uniform distribution is the midpoint of the range: E(X)=2a+b
Variance of a uniform distribution is Var(X)=12(b−a)2
The Normal Distribution
Normal distribution denoted as N(μ,σ2), where μ is the mean and σ2 is the variance
PDF of a normal distribution is a bell-shaped curve: f(x)=σ2π1e−2σ2(x−μ)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∼N(0,1)
Any normal distribution can be standardized using the z-score: Z=σX−μ
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≤X≤b)=∫abf(x)dx
For uniform distributions, probabilities can be calculated using the CDF: P(X≤x)=F(x)=b−ax−a for a≤x≤b
For normal distributions, probabilities are often calculated using z-scores and standard normal tables
First, standardize the variable: Z=σX−μ
Then, use tables or software to find the probability: P(X≤x)=P(Z≤σx−μ)
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∼U(2,7) is between 3 and 5
P(3≤X≤5)=7−25−3=52
Find the 90th percentile of a normal distribution with mean 50 and standard deviation 10
90th percentile corresponds to z=1.28, so x=μ+zσ=50+1.28(10)=62.8
Determine the expected value and variance of a uniform distribution U(−3,6)
E(X)=2−3+6=1.5 and Var(X)=12(6−(−3))2=1281=6.75
Calculate the probability that a standard normal random variable is greater than 1.5