Solution Found!

The Exponential is the analog of the Geometric in continuous time. This problem explores

Chapter 5, Problem 45

(choose chapter or problem)

Get Unlimited 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)

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

Add to cart


Study Tools You Might Need

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back