A bridge has caused a lot of accidents. A rough statistics shows for every 100,000 vehicles passing by the bridge, 3 of them will crash. (a) There are approximately 250,000 vehicles passing by the bridge in a month. Let X be the number of crashed vehicles. What distribution does X follow (that is what kind of random variable is X)? State the assumptions. Use it to compute the expected value of X and the probability of having ≥ 2 car crashes. (b) By law of rare events, X is approximated by a Poisson random variable. What is the intensity? Use this to compute the probability of having ≥ 2 car crashes. Compare the result with (a). (c) For every crashed vehicle, with 0.25 probability the damage is classified as "severe." Let Y be the number of severely crashed car within two months. Find the probability of Y ≥ 3 using Poisson approximation.
A bridge has caused a lot of accidents. A rough statistics shows for every 100,000 vehicles passing by the bridge, 3 of them will crash. (a) There are approximately 250,000 vehicles passing by the bridge in a month. Let X be the number of crashed vehicles. What distribution does X follow (that is what kind of random variable is X)? State the assumptions. Use it to compute the expected value of X and the probability of having ≥ 2 car crashes. (b) By law of rare events, X is approximated by a Poisson random variable. What is the intensity? Use this to compute the probability of having ≥ 2 car crashes. Compare the result with (a). (c) For every crashed vehicle, with 0.25 probability the damage is classified as "severe." Let Y be the number of severely crashed car within two months. Find the probability of Y ≥ 3 using Poisson approximation.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:A bridge has caused a lot of accidents. A rough statistics shows for every 100,000 vehicles
passing by the bridge, 3 of them will crash.
(a) There are approximately 250,000 vehicles passing by the bridge in a month. Let X be the
number of crashed vehicles. What distribution does X follow (that is what kind of random
variable is X)? State the assumptions. Use it to compute the expected value of X and the
probability of having ≥ 2 car crashes.
(b) By law of rare events, X is approximated by a Poisson random variable. What is the
intensity? Use this to compute the probability of having ≥ 2 car crashes. Compare the result
with (a).
(c) For every crashed vehicle, with 0.25 probability the damage is classified as "severe." Let
Y be the number of severely crashed car within two months. Find the probability of Y ≥ 3
using Poisson approximation.
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