Lecture#16.pdf - Contents 16 Gamma Distribution 1 16.1...

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Contents 16 Gamma Distribution 1 16.1 Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 16.2 Expectation & Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 16.3 Gamma in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 16.4 Related Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 16.4.1 Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 16.4.2 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 16.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 16 Gamma Distribution 16.1 Probability Density Function Introduction We need a library of continuous random variables to characterize the distributions of continuous random outcomes visible in everyday life. So far, we have defined the Uniform and Normal distributions, but they have limited applicability in certain situations. Consider the random variable describing the time it takes for some random event to happen (e.g. bus to arrive, service at the check-out to finish, Kentucky Derby winner to finish the race, etc.). It is not reasonable to hypothesize a uniform distribution for this “wait time” because there is not concrete lower and upper limit; in addition, we expect the distribution to be notably peaked at the most likely times. Nor is it reasonable to use a normal distribution, because the pdf puts positive, albeit possibly very small, probability on negative numbers, and wait times can never be negative. We need another kind of distribution useful for random variables with range in the positive real line. Gamma and its special cases, exponential and chi-square, come to the rescue. Gamma pdf Definition : Gamma distribution The random variable Y Gamma ( α, β ) is said to have a Gamma distribution with shape pa- rameter α and scale parameter β
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