10/22/2012

Exponential and Gamma distributions

• Exponential - p.d.f, c.d.f, m.g.f, mean, variance, memoryless property
• Note: An exponential distribution is often used in a practical problem to represent the distribution of the time that elapses before the occurence of some event. If the events being considered occur in accordance with a Poisson process, then both the waiting time until the event occurs and the time between any two successive events will have exponential distributions.
• Example - Phone calls at an exchange arrive at the rate of 8 per minute, according to a Poisson process. Find the probability that the time, T, from any particular time to the next call exceeds 10 seconds.
• Memoryless property of the Exponential Distribution
• 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

Please read sections 6.2-6.3 for class on Wednesday.