pt 1: Normal, Uniform, Exponential

Normal Distribution

  • Under certain conditions, the normal distribution can replace the binomial distribution.
  • It is a foundation for many statistical models.

Real-world Examples

  • Door Handle Germs: Most people touch the middle of a door, so germs are more likely to be there. If you know the distribution you can make predictions.
  • Manufacturing: Stronger materials are placed in the middle because that's where the most force is applied.

Deviations

  • If data deviates from the normal distribution, other shapes arise.

Formula for Normal Distribution

  • The formula uses xx (x variable), pipi, σ\sigma (sigma), ee (exponential function), and μ\mu (mu).
  • f(x)=12πσ2e−12(x−μ)2σ2f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} \frac{(x-\mu)^2}{\sigma^2}}
  • This is a density function.

Density Function Properties

  • The area under the density function curve is always equal to one.
  • Probability is related to area.
  • Calculating area gives probability.

Symmetry

  • The normal distribution is symmetric.
  • The area to the left and right of the mean is 0.5 each.

Fusion Peak

  • The peak occurs when x=μx = \mu.

Range of Data

  • Data (x) can range from negative infinity to positive infinity.
  • Can deal with negative and positive numbers, both large.

Continuous Distribution

  • Unlike discrete distributions with spikes, the normal distribution has an infinite number of spikes, forming a continuous line.
  • The curve never touches the x-axis (goes on to infinity).
  • The normal distribution curve extends to infinity and never touches the x-axis. This is a key characteristic of the distribution.

Gaussian Distribution

  • The normal distribution is also known as the Gaussian distribution, named after Gauss.

Using Tables

  • We will use tables to find areas instead of directly using the formula.

Uniform Distribution

Characteristics

  • It has a beginning (a) and an end (b).
  • f(x)=1b−af(x) = \frac{1}{b-a}
  • x is always between a and b.
  • It's a flat histogram.

Area Calculation

  • Area = height * width
  • Total area is equal to one.

Mean

  • The mean is in the middle.
  • μ=b+a2\mu = \frac{b + a}{2}

Parameters

  • For normal distribution: mean and variance.
  • For uniform distribution: a and b.

Exponential Distribution

Association

  • Associated with lifetime.
  • Initially, there's a long lifetime, but most products have a shorter lifetime.

Shape

  • Often drawn decreasing, but can also go up and down.

Parameter

  • Uses the same parameter as the Poisson distribution: lambda (λ\lambda).

Formula

  • f(x)=λe−λxf(x) = \lambda e^{-\lambda x}
  • xx is bigger than or equal to zero.
  • Anywhere else, the density is zero.

Area

  • The area under the curve is equal to one.
  • ∫0∞λe−λxdx=1\int_{0}^{\infty} \lambda e^{-\lambda x} dx = 1

Symmetry

  • Not symmetric.
  • Mean, median, and mode are different.

Parameter

  • We say xx is an exponential distribution with parameter λ\lambda.
  • λ\lambda is similar to the Poisson distribution: number of phone calls in an hour.
  • Modeling the time of the phone call, not the number of calls.