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== Problem 1: Ceiling of an Exponential == | == Problem 1: Ceiling of an Exponential == | ||
<math>X</math> is an exponential random variable with paramter <math>\lambda</math>. <math>Y = \mathrm{ceil}(X)</math>, where the ceiling function <math>\mathrm{ceil}(\cdot)</math> rounds its argument up to the closest integer, i.e.: | <math>X</math> is an exponential random variable with paramter <math>\lambda</math>. <math>Y = \mathrm{ceil}(X)</math>, where the ceiling function <math>\mathrm{ceil}(\cdot)</math> rounds its argument up to the closest integer, i.e.: | ||
| + | |||
| + | <math>\mathrm{ceil}(a)</math> = <math>a</math> if <math>a</math> is an integer | ||
| + | = the smallest integer bigger than <math>a</math> if <math>a</math> is not an integer | ||
What is the PMF of <math>Y</math>? Is it one of the common random variables? (Hint: for all <math>k</math>, find the quantity <math>P(Y > k)</math>. Then find the PMF) | What is the PMF of <math>Y</math>? Is it one of the common random variables? (Hint: for all <math>k</math>, find the quantity <math>P(Y > k)</math>. Then find the PMF) | ||
Revision as of 07:18, 8 October 2008
Instructions
Homework 6 can be downloaded here on the ECE 302 course website.
Problem 1: Ceiling of an Exponential
$ X $ is an exponential random variable with paramter $ \lambda $. $ Y = \mathrm{ceil}(X) $, where the ceiling function $ \mathrm{ceil}(\cdot) $ rounds its argument up to the closest integer, i.e.:
$ \mathrm{ceil}(a) $ = $ a $ if $ a $ is an integer = the smallest integer bigger than $ a $ if $ a $ is not an integer
What is the PMF of $ Y $? Is it one of the common random variables? (Hint: for all $ k $, find the quantity $ P(Y > k) $. Then find the PMF)
