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MANE 3332.04

Lecture 11, February 26

Agenda

  • Complete Chapter 3
    • Topics: Poisson distribution
  • Start Chapter 4
    • Topics: Continuous RV, pdf, CDF, mean and variance, Uniform distribution
  • New: Poisson Practice Problems (assigned 2/26/2025, due 3/3/2025 11:59pm)

Handouts

Schedule

  • February 26: Poisson Practice Problems, Chapter 4: Continuous RV, pdf, mean and variance, Uniform distribution
  • March 3: Standard normal distribution
  • March 5: Normal distribution
  • March 10: Exponential and Weibull distributions
  • March 12: Mid-term review/Tech Report Assignment
  • March 17: Spring Break
  • March 19: Spring Break
  • March 24: Chapter 5 or midterm
  • March 26: midterm or Chapter 5

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Poisson Process

Assume that the events occur at random throughout the interval. If the interval can be partitioned into subintervals of small enough length such that

  1. The probability of more than one count in a subinterval is zero

  2. The probability of one count in a subinterval is the same for all subintervals and proportional to the length of the subinterval, and

  3. The count in each subinterval is independent of other subintervals, the random experiment is called a Poisson process

Poisson Distribution

If the mean number of counts in the interval is \(\lambda>0\), the random variable \(X\) that equals the number of counts in the interval has a Poisson distribution with parameter \(\lambda\)

  • The Poisson PMF is
\[ f(x)=\frac{e^{-\lambda}\lambda^x}{x!},\; x=0,1,2,\ldots \]
  • The mean of a Poisson random variable is
\[ E(X)=\mu=\lambda \]
  • The variance of a Poisson random variable is
\[ V(X)=\sigma^2=\lambda \]

Poisson Practice Problems

Poisson Example

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