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If gene frequencies are in Hardy-Weinberg equilibrium, the genotypes AA, Aa and aa occur with probabilities (1–0)², 20(1-0) and 0². Suppose that on haptoglobin type in a sample of n people, n₁ individuals have genotypes AA, n₂ individuals have genotypes Aa and n3 individuals have genotypes aa: AA n1 Haptoglobin Type Aa n2 aa n3 Thus, (n₁, n2, n3) is multinomial with parameters n and (1 - 0)², 20(1-0) and 0². (a) Write down the likelihood function L(0) of 0. (b) Assume that the prior for is uniform between 0 and 1. Find the posterior

If gene frequencies are in Hardy-Weinberg equilibrium, the genotypes AA, Aa and aa occur with probabilities (1–0)², 20(1-0) and 0². Suppose that on haptoglobin type in a sample of n people, n₁ individuals have genotypes AA, n₂ individuals have genotypes Aa and n3 individuals have genotypes aa: AA n1 Haptoglobin Type Aa n2 aa n3 Thus, (n₁, n2, n3) is multinomial with parameters n and (1 - 0)², 20(1-0) and 0². (a) Write down the likelihood function L(0) of 0. (b) Assume that the prior for is uniform between 0 and 1. Find the posterior

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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a) If gene frequencies are in Hardy-Weinberg equilibrium, the genotypes AA, Aa and aa occur with probabilities (1-0)², 20(1-0) and 0². Suppose that on haptoglobin type in a sample of n people, n₁ individuals have genotypes AA, n₂ individuals have genotypes Aa and n3 individuals have genotypes aa: b) AA n1 Haptoglobin Type Aa n2 Solution: Thus, (n₁, n2, n3) is multinomial with parameters n and (1 - 0)², 20(1-0) and 0². (a) Write down the likelihood function L(0) of 0. (b) Assume that the prior for is uniform between 0 and 1. Find the posterior distribution of given data n₁, n2 and n3. Is this a known distribution? (c) Find the Jeffrey's prior for 0. f(e\n₁, n₂, n3) 0 x L(0)=P(N₁ = n₁, №₂ = n₂, N3 = 13) n! n₁!· n₂! n3! x x aa n3 (1 π₁ (0)=√(1-0)² 2n 0)²″ ¹(20(1 – 0))^²(0)²n ³3 n! n₁!·n₂! n3! n! n₁!·n₂! -n₂! 22xn! n₁!·n₂! n3! 2n 2n (1 – 0)²″ ¹(20(1 – 0))^²(0)²n 3 (1-0) (1 -0 3 ¹(2″ ²0″²(1 – 0)^²) (0)²¹ ³ 2n₁+n₂ n₂+2n3 The Jeffreys prior for the parameter of the continuous probability density function is 2n₁ n₂(-20²+20-1) n3 0²(1-0)² 0² +
Transcribed Image Text:a) If gene frequencies are in Hardy-Weinberg equilibrium, the genotypes AA, Aa and aa occur with probabilities (1-0)², 20(1-0) and 0². Suppose that on haptoglobin type in a sample of n people, n₁ individuals have genotypes AA, n₂ individuals have genotypes Aa and n3 individuals have genotypes aa: b) AA n1 Haptoglobin Type Aa n2 Solution: Thus, (n₁, n2, n3) is multinomial with parameters n and (1 - 0)², 20(1-0) and 0². (a) Write down the likelihood function L(0) of 0. (b) Assume that the prior for is uniform between 0 and 1. Find the posterior distribution of given data n₁, n2 and n3. Is this a known distribution? (c) Find the Jeffrey's prior for 0. f(e\n₁, n₂, n3) 0 x L(0)=P(N₁ = n₁, №₂ = n₂, N3 = 13) n! n₁!· n₂! n3! x x aa n3 (1 π₁ (0)=√(1-0)² 2n 0)²″ ¹(20(1 – 0))^²(0)²n ³3 n! n₁!·n₂! n3! n! n₁!·n₂! -n₂! 22xn! n₁!·n₂! n3! 2n 2n (1 – 0)²″ ¹(20(1 – 0))^²(0)²n 3 (1-0) (1 -0 3 ¹(2″ ²0″²(1 – 0)^²) (0)²¹ ³ 2n₁+n₂ n₂+2n3 The Jeffreys prior for the parameter of the continuous probability density function is 2n₁ n₂(-20²+20-1) n3 0²(1-0)² 0² +
Continue with Question 3 and answer the following: (a) Use the data on the haptoglobin type and compute the MLE Ô for 0 and I(Ô) the Fisher information evaluated at the MLE. (b) Use the normal approximation for posterior distribution to find a 95% credible interval for 0.
Transcribed Image Text:Continue with Question 3 and answer the following: (a) Use the data on the haptoglobin type and compute the MLE Ô for 0 and I(Ô) the Fisher information evaluated at the MLE. (b) Use the normal approximation for posterior distribution to find a 95% credible interval for 0.
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