11/1/2023

Chapter 7 - Continuous Distributions continued

See Chapter7 - Gamma distributions.pdf in our Google Drive folder

• Gamma Function
• Properties of the gamma function
• Relating the gamma function and the standard normal distribution
• Gamma Distribution
• Properties of the Gamma Distribution
  • Find the k-th non-central moment for the gamma distribution.
  • Use your result for the k-th non-central moment to find the mean and variance of a gamma random variable.
• Moment generating function
• The exponential distribution is a special case of the gamma distribution
• Calculating probabilities, finding percentiles, and generating random observations for normal distributions
  • dgamma(k, shape, scale), pgamma(k, shape, scale), qgamma(k, shape, scale), and rgamma(n, shape, scale) with RStudio
• Example - If a company employees n salespersons, it gross sales in thousands of dollars may be regarded as a random variable having a gamma distribution with alpha=80sqrt(n) and beta=1/2. If the sales cost is $8000 per person, how many salespersons should the company employ to maximize the expected profit?
• Beta Distribution
• mean and variance
• Moment generating function and other properties
• Calculating probabilities, finding percentiles, and generating random observations for normal distributions
  • dbeta(k, shape1, shape2), pbeta(k, shape1, shape2), qbeta(k, shape1, shape2), and rbeta(n, shape1, shape2) with RStudio

We will have a problem session on Friday. Please complete your reading of Chapter 7 for class on Monday.