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Survival Analysis with Exponential and Gamma Distributions

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Learn about exponential and gamma distributions for modeling waiting times and Poisson processes. Calculate probabilities and expected values. Ideal for studying radioactive decay and inter-arrival times.

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Survival Analysis with Exponential and Gamma Distributions

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  1. Exponential Distribution • The RV T has an exponential distribution with rate l (for l > 0) if T has the probability density • The mean and standard deviation of T are

  2. Exponential Distribution • The exponential distribution is used to model waiting times for the occurrence of some event (death, failure, mutation, radioactive decay, etc.). • The continuous analog of the geometric distribution. • Models the successive inter-arrival times of a Poisson process in time.

  3. Suppose a particular kind of radioactive atom has a half-life of 2 years. Find • The probability that an atom of this type survives at least 5 years. • The time at which the expected number of atoms is 10% of the original amount.

  4. Gamma Distribution • If Tr is the time of the rth arrival after t = 0 in a Poisson process with rate l, then Tr has the gamma (r, l) distribution with probability density • The mean and standard deviation of Tr are

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