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(Sonar) A scientist is trying to determine the depth of the sea at a certain location. She knows that it must be deeper than 5 km but that is all she knows. To capture this uncertainty she models the depth as uniformly distributed between 5 and 10 km. In order to measure the depth she uses sonar, taking 2 measurements, which we model as two random variables 3₁ and 3₂. If the depth is equal to x then each sonar measurement is uniformly distributed between x 0.25 and x +0.25. The two measurements are conditionally independent given the depth. (a) Compute and sketch the pdf of the first sonar measurement š₁. 2 ) Compute the conditional pdf of the depth conditioned on the measurements being equal to 7 km and 7.1 km. ) Compute the joint pdf of the two sonar measurements š₁ and 52. Are the two measurements independent? Justify your answer mathematically and explain it in- tuitively.

(Sonar) A scientist is trying to determine the depth of the sea at a certain location. She knows that it must be deeper than 5 km but that is all she knows. To capture this uncertainty she models the depth as uniformly distributed between 5 and 10 km. In order to measure the depth she uses sonar, taking 2 measurements, which we model as two random variables 3₁ and 3₂. If the depth is equal to x then each sonar measurement is uniformly distributed between x 0.25 and x +0.25. The two measurements are conditionally independent given the depth. (a) Compute and sketch the pdf of the first sonar measurement š₁. 2 ) Compute the conditional pdf of the depth conditioned on the measurements being equal to 7 km and 7.1 km. ) Compute the joint pdf of the two sonar measurements š₁ and 52. Are the two measurements independent? Justify your answer mathematically and explain it in- tuitively.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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(Sonar) A scientist is trying to determine the depth of the sea at a certain location. She knows that it must be deeper than 5 km but that is all she knows. To capture this uncertainty she models the depth as uniformly distributed between 5 and 10 km. In order to measure the depth she uses sonar, taking 2 measurements, which we model as two random variables š1 and 52. If the depth is equal to x then each sonar measurement is uniformly distributed between x conditionally independent given the depth. 0.25 and + 0.25. The two measurements are (a) Compute and sketch the pdf of the first sonar measurement š1. 2 b) Compute the conditional pdf of the depth conditioned on the measurements being equal to 7 km and 7.1 km. c) Compute the joint pdf of the two sonar measurements ši and 52. Are the two measurements independent? Justify your answer mathematically and explain it in- tuitively.
Transcribed Image Text:(Sonar) A scientist is trying to determine the depth of the sea at a certain location. She knows that it must be deeper than 5 km but that is all she knows. To capture this uncertainty she models the depth as uniformly distributed between 5 and 10 km. In order to measure the depth she uses sonar, taking 2 measurements, which we model as two random variables š1 and 52. If the depth is equal to x then each sonar measurement is uniformly distributed between x conditionally independent given the depth. 0.25 and + 0.25. The two measurements are (a) Compute and sketch the pdf of the first sonar measurement š1. 2 b) Compute the conditional pdf of the depth conditioned on the measurements being equal to 7 km and 7.1 km. c) Compute the joint pdf of the two sonar measurements ši and 52. Are the two measurements independent? Justify your answer mathematically and explain it in- tuitively.
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