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Exercise 1) Your friend transmits an unknown value to you over a noisy channel. The noise is normally distributed with mean 0 and a known variance 4, so the value x that you receive is modeled by N(0,4). Based on previous communications, your prior on is N (5,9). (a) Suppose your friend transmits a value to you that you receive as x = 6. Show that the posterior pdf for is N(74/13, 36/13). For this problem, you need to derive the posterior by carrying out the calculations from scratch. = (b) Suppose your friend transmits the same value to you n = 4 times. You receive these signals plus noise as x₁, ..., 4 with sample mean 6. Using the same prior and known variance o² as in part (a), show that the posterior on is N(5.9,0.9). Plot the posterior and posterior on the same graph. Describe how the data changes your belief about the true value of 0. For this question, you may use the normal updating formulas.

Exercise 1) Your friend transmits an unknown value to you over a noisy channel. The noise is normally distributed with mean 0 and a known variance 4, so the value x that you receive is modeled by N(0,4). Based on previous communications, your prior on is N (5,9). (a) Suppose your friend transmits a value to you that you receive as x = 6. Show that the posterior pdf for is N(74/13, 36/13). For this problem, you need to derive the posterior by carrying out the calculations from scratch. = (b) Suppose your friend transmits the same value to you n = 4 times. You receive these signals plus noise as x₁, ..., 4 with sample mean 6. Using the same prior and known variance o² as in part (a), show that the posterior on is N(5.9,0.9). Plot the posterior and posterior on the same graph. Describe how the data changes your belief about the true value of 0. For this question, you may use the normal updating formulas.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 5E
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Exercise 1) Your friend transmits an unknown value to you over a noisy channel. The noise is normally distributed with mean 0 and a known variance 4, so the value x that you receive is modeled by N(0,4). Based on previous communications, your prior on is N (5,9). (a) Suppose your friend transmits a value to you that you receive as x = 6. Show that the posterior pdf for is N(74/13, 36/13). For this problem, you need to derive the posterior by carrying out the calculations from scratch. (b) Suppose your friend transmits the same value 0 to you n = 4 times. You receive these signals plus noise as x₁, ..., 4 with sample mean x = 6. Using the same prior and known variance o² as in part (a), show that the posterior on 6 is N(5.9,0.9). Plot the posterior and posterior on the same graph. Describe how the data changes your belief about the true value of 0. For this question, you may use the normal updating formulas.
Transcribed Image Text:Exercise 1) Your friend transmits an unknown value to you over a noisy channel. The noise is normally distributed with mean 0 and a known variance 4, so the value x that you receive is modeled by N(0,4). Based on previous communications, your prior on is N (5,9). (a) Suppose your friend transmits a value to you that you receive as x = 6. Show that the posterior pdf for is N(74/13, 36/13). For this problem, you need to derive the posterior by carrying out the calculations from scratch. (b) Suppose your friend transmits the same value 0 to you n = 4 times. You receive these signals plus noise as x₁, ..., 4 with sample mean x = 6. Using the same prior and known variance o² as in part (a), show that the posterior on 6 is N(5.9,0.9). Plot the posterior and posterior on the same graph. Describe how the data changes your belief about the true value of 0. For this question, you may use the normal updating formulas.
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