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Exponential and Chi-Square Random Variables

Exponential and Chi-Square Random Variables. Recall Poisson R. V. In a fixed time interval of length T , if there are an average of l arrivals, then “number of arrivals” has a Poisson distribution:. where y = 0, 1, 2, ….

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Exponential and Chi-Square Random Variables

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  1. Exponential and Chi-Square Random Variables

  2. Recall Poisson R. V. • In a fixed time interval of length T, if there are an average of larrivals, then “number of arrivals” has a Poisson distribution: where y = 0, 1, 2, … Similarly, given the average number of arrivals per unit time, say in l*arrivals per minute…

  3. Poisson R. V. • …then in T minutes, we expect l*T arrivals, and so “number of arrivals” in T minutes has a Poisson distribution: where y = 0, 1, 2, … Consider the time between arrivals. That is, consider the “inter-arrival times”.

  4. 0.2 arrivals per minute • If customers arrive at an average of 0.2 arrivals per minute, find the probability of 3 arrivals in a 10-minute period. • Note l*T = 2 arrivals and so • Find the probability of no arrivals in the 10-minute period.

  5. Time till arrival? • Consider W, the time until the first arrival. Number of customers t T • W is a continuous random variable. What can we say about its probability distribution?

  6. Inter-arrival Distribution • Note that F(w) = P(W < w) = 1 – P(W > w) Number of customers w t • Time of first arrival W > w implies zero arrivals have occurred in the interval (0, w). • Don’t we already know this probability?

  7. Inter-arrival times • If the average arrivals per unit time equals l, the probability that zero arrivals have occurred in the interval (0, w) is given by the Poisson distribution F(w) = P(W < w) = 1 – P(W > w) Sometimes writtenwhere b = 1/ l is the average inter-arrival time(e.g., “minutes per arrival”).

  8. Exponential Distribution • A continuous random variable W whose distribution and density functions are given by and is said to have an exponential distribution with parameter (“average”) b .

  9. Exponential Random Variables • Typical exponential random variables may include: • Time between arrivals (inter-arrival times) • Service time at a server (e.g., CPU, I/O device, or a communication channel) in a queueing network. • Time to failure (“lifetime”) of a component.

  10. 0.2 arrivals per minute Distributions for W, time till first arrival: ( using integration-by-parts ) As expected, since average time is 1/0.2 = 5 minutes/arrival.

  11. Exponential mean, variance • If W is an exponential random variable with parameter b, the expected value and variance for W are given by Also, note that

  12. Problem 4.74 • Air samples in a city have CO concentrations that are exponentially distributed with mean 3.6 ppm. • For a randomly selected sample, find the probability the concentration exceeds 9 ppm. • If the city manages its traffic such that the mean CO concentration is reduced to 2.5 ppm, then what is the probability a sample exceeds 9 ppm?

  13. Problem 4.82 • The lifetime of a component is exponentially distributed with an average b = 100 hours/failure. • Three of these components operate independently in a piece of equipment and the equipment fails if at least 2 of the components fail. • Find the probability the equipment operates for at least 200 hours without failure.

  14. Density Curves Exponential distributions for some various rates l.

  15. Memoryless • Note P(W > w) = 1 – P(W < w) = 1 – (1 – e-lw) = e-lw • Consider the conditional probability P(W > a + b | W > a ) = P(W > a + b)/P(W > a) • We find that The only continuous memoryless random variable.

  16. Gamma Distribution • The exponential distribution is a special case of the more general gamma distribution: where the gamma function is For the exponential, choose a = 1 and note G(1) = 1.

  17. Gamma Density Curves Gamma function facts:

  18. Exponential mean, variance • If Y has a gamma distribution with parameters a and b, the expected value and variance for Y are given by In the case of a = 1, the values for the exponential distribution result.

  19. Deriving the Mean • By definition of the density function • Since this holds for any a > 0, note that

  20. Deriving the Mean • Now, consider the expected value

  21. Problem 4.88 • Find E(Y) and V(Y) by inspectiongiven that

  22. Chi-Square Distribution • As another special case of the gamma distribution, consider letting a = v/2 and b = 2, for some positive integer v. This defines the Chi-square distribution. Note the mean and variance are given by

  23. Statistical Testing • For a sample of size n, with variance s2. • To compare against a given value s02 • We find that the ratio (n – 1)s2/s02 has the chi-square distribution with v = n – 1 degrees of freedom. • Develop and test the “null hypothesis” based on the chi-square probability distribution. Get the details on hypothesis testing in MAT 432 in the Spring!

  24. Practice Problems • Work problems:4.69, 4.71, 4.73, 4.77, 4.78, 4.81, 4.83

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