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The Exponential is the analog of the Geometric in continuous time. This problem explores
Chapter 5, Problem 45(choose chapter or problem)
The Exponential is the analog of the Geometric in continuous time. This problem explores the connection between Exponential and Geometric in more detail, asking what happens to a Geometric in a limit where the Bernoulli trials are performed faster and faster but with smaller and smaller success probabilities. Suppose that Bernoulli trials are being performed in continuous time; rather than only thinking about first trial, second trial, etc., imagine that the trials take place at points on a timeline. Assume that the trials are at regularly spaced times 0, t, 2t, . . . , where t is a small positive number. Let the probability of success of each trial be t, where is a positive constant. Let G be the number of failures before the first success (in discrete time), and T be the time of the first success (in continuous time). (a) Find a simple equation relating G to T. Hint: Draw a timeline and try out a simple example. (b) Find the CDF of T. Hint: First find P(T >t). (c) Show that as t ! 0, the CDF of T converges to the Expo() CDF, evaluating all the CDFs at a fixed t 0. Hint: Use the compound interest limit (see the math appendix)
Questions & Answers
QUESTION:
The Exponential is the analog of the Geometric in continuous time. This problem explores the connection between Exponential and Geometric in more detail, asking what happens to a Geometric in a limit where the Bernoulli trials are performed faster and faster but with smaller and smaller success probabilities. Suppose that Bernoulli trials are being performed in continuous time; rather than only thinking about first trial, second trial, etc., imagine that the trials take place at points on a timeline. Assume that the trials are at regularly spaced times 0, t, 2t, . . . , where t is a small positive number. Let the probability of success of each trial be t, where is a positive constant. Let G be the number of failures before the first success (in discrete time), and T be the time of the first success (in continuous time). (a) Find a simple equation relating G to T. Hint: Draw a timeline and try out a simple example. (b) Find the CDF of T. Hint: First find P(T >t). (c) Show that as t ! 0, the CDF of T converges to the Expo() CDF, evaluating all the CDFs at a fixed t 0. Hint: Use the compound interest limit (see the math appendix)
ANSWER:Step 1 of 3
Given that G is the number of failures before the first success and T be the time of first success.
The trials are at regularly spaced times where is a small positive number.
Since G is the number of failures before the first success, therefore, before G, there are trials and time consumed for is . After that, we take another for trail and after that we have the first success.
So, total time consumed is
Hence, the relation between T and G is