11/8/2023

Chapter 7 - Continuous Distributions

• Pareto distribution - will be discussed in a student presentation
• General discussion about the form and derivation of the pdf
• Calculating probabilities, finding percentiles, and generating random observations for normal distributions
  • You need to install the EnvStats package
  • dpareto(k, df), ppareto(k, df), qpareto(p, df), and rpareto(n, df) with RStudio

Chapter 8 - Densities of functions of random variables

• Cumulative distribution function technique - incorporated into Chapter 6, but covered Section 8.1
  • Cauchy distribution
    • You need to install the stats package
    • dcauchy(k, location, scale), pcauchy(k, location, scale), qcauchy(p, location, scale), and rcauchy(n, location, scale) with RStudio
  • Weibull distribution
    • You need to install the stats package
    • dweibull(k, shape, scale), pweibull(k, shape, scale), qweibull(p, shape, scale), and rweibull(n, shape, scale) with RStudio
• Simulating a continuous random variable - see Chapter 8 - Simulation in the folder !Class Material on Google Drive
  • Probability integral transformation theorem and the inverse transform method
• Method of transformations - also called the Jacobian technique - incorporated into Chapter 6
• Order Statistics - see Chapter 8 - Order Statistics in the folder !Class Material on Google Drive
  • We have already seen exercises with the minimum and the maximum, but now we generalize that work to find the general form of the distributions of the minimum, maximum, and any other order statistic.
• Convolution
  • We have seen how to find the distributions of sums using the m.g.f., but this technique provides another method to find the distribution of a sum.
• Geometric Probability - lots of cool examples like Buffon's needle problem in Example 8.23.
• Transformations of two random variables

Class Activity

• Use the converse of the PIT Theorem to generate random variables from an exponential distribution with lamda=1.
• Simulate the distribution of the minimum and maximum for a set of random variables from a normal distribution with mean 75 and standard deviation 10. Are the shapes of your two distributions the same?

We will have problem sessions on Monday and Friday and then review for exam #2.