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The human IQ is normally distributed with a mean of 100 points and a standard deviation of 15 points. A test measures the IQ with a standard deviation of 5 points. If someone scores 100 (or 150, 200) points on this test, what is a reasonable Bayesian estimate of his/her real IQ? Specifically, consider X₁N(0, 52) with a prior distribution on 0; being 0%; ~ N(100, 15²). (a) Find the posterior distribution of 0; given X₂. (b) What's the posterior expectation, E(X), for X1 = 100, X2 = 150 and X3 = 200, respectively? Do Bayesians appear to trust seemingly smart people or not? [Think: What's the statistical intuition behind their apparent trust or distrust?

The human IQ is normally distributed with a mean of 100 points and a standard deviation of 15 points. A test measures the IQ with a standard deviation of 5 points. If someone scores 100 (or 150, 200) points on this test, what is a reasonable Bayesian estimate of his/her real IQ? Specifically, consider X₁N(0, 52) with a prior distribution on 0; being 0%; ~ N(100, 15²). (a) Find the posterior distribution of 0; given X₂. (b) What's the posterior expectation, E(X), for X1 = 100, X2 = 150 and X3 = 200, respectively? Do Bayesians appear to trust seemingly smart people or not? [Think: What's the statistical intuition behind their apparent trust or distrust?

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.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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I need help solving this example by using the posterior formula and computing the integral. The trick is to factor things out in such a way the integral is an integral of a distribution and thus it just becomes 1 with a constant factored out, but i can't do it.  

The human IQ is normally distributed with a mean of 100 points and a standard deviation of 15 points. A test measures the IQ with a standard deviation of 5 points. If someone scores 100 (or 150, 200) points on this test, what is a reasonable Bayesian estimate of his/her real IQ? Specifically, consider Xi ~ N(0i, 5²) with a prior distribution on ¤¿ being ; ~ N(100, 15²). (a) Find the posterior distribution of 0; given X₁. (b) What's the posterior expectation, E(0¿|Xi), for X1 = - 100, X2 150 and X3 = 200, respectively? Do Bayesians appear to trust seemingly smart people or not? [Think: What's the statistical intuition behind their apparent trust or distrust?]
Transcribed Image Text:The human IQ is normally distributed with a mean of 100 points and a standard deviation of 15 points. A test measures the IQ with a standard deviation of 5 points. If someone scores 100 (or 150, 200) points on this test, what is a reasonable Bayesian estimate of his/her real IQ? Specifically, consider Xi ~ N(0i, 5²) with a prior distribution on ¤¿ being ; ~ N(100, 15²). (a) Find the posterior distribution of 0; given X₁. (b) What's the posterior expectation, E(0¿|Xi), for X1 = - 100, X2 150 and X3 = 200, respectively? Do Bayesians appear to trust seemingly smart people or not? [Think: What's the statistical intuition behind their apparent trust or distrust?]
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