# Using this and independence we have p mins t t p s

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Unformatted text preview: s will die out. 1.77. Consider a branching process as deﬁned in Example 7.2, in which each family has a number of children that follows a shifted geometric distribution: pk = p(1 p)k for k 0, which counts the number of failures before the ﬁrst success when success has probability p. Compute the probability that starting from one individual the chain will be absorbed at 0. Chapter 2 Poisson Processes 2.1 Exponential Distribution To prepare for our discussion of the Poisson process, we need to recall the deﬁnition and some of the basic properties of the exponential distribution. A random variable T is said to have an exponential distribution with rate , or T = exponential( ), if P (T t) = 1 t e for all t 0 (2.1) Here we have described the distribution by giving the distribution function F (t) = P (T t). We can also write the deﬁnition in terms of the density function fT (t) which is the derivative of the distribution function. ( e t for t 0 fT (t) = (2.2) 0 for t &lt; 0 Integrating by parts with f (t) = t and g 0 (t) = e t , Z Z...
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