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(a) Let X₁, X2,..., Xn be random variables from a normal distribution with unknown mean μ and unknown variance o². We are interested in finding the maximum likelihood estimates of u and o². Now let û and 2 be the maximum likelihood estimates for u and o2. The probability density function of X, is given by 1 f(x₁ ; μ, 0²): -22 (Ti-μ)² = 2πσ2 for - <μ<∞, 0 < o² < ∞ and i = 1, 2, ..., n. Prove that Σ₁=1 Xi Σ=1(xi - μ)2 μ = and 2 = n n

(a) Let X₁, X2,..., Xn be random variables from a normal distribution with unknown mean μ and unknown variance o². We are interested in finding the maximum likelihood estimates of u and o². Now let û and 2 be the maximum likelihood estimates for u and o2. The probability density function of X, is given by 1 f(x₁ ; μ, 0²): -22 (Ti-μ)² = 2πσ2 for - <μ<∞, 0 < o² < ∞ and i = 1, 2, ..., n. Prove that Σ₁=1 Xi Σ=1(xi - μ)2 μ = and 2 = n n

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.CR: Chapter 13 Review
Problem 7CR
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(a) Let X₁, X2,..., Xn be random variables from a normal distribution with unknown mean μ and unknown variance o². We are interested in finding the maximum likelihood estimates of u and o². Now let û and 2 be the maximum likelihood estimates for u and o2. The probability density function of X, is given by 1 f(x₁ ; μ, 0²): -22 (Ti-μ)² = 2πσ2 for - <μ<∞, 0 < o² < ∞ and i = 1, 2, ..., n. Prove that Σ₁=1 Xi Σ=1(xi - μ)2 μ = and 2 = n n
Transcribed Image Text:(a) Let X₁, X2,..., Xn be random variables from a normal distribution with unknown mean μ and unknown variance o². We are interested in finding the maximum likelihood estimates of u and o². Now let û and 2 be the maximum likelihood estimates for u and o2. The probability density function of X, is given by 1 f(x₁ ; μ, 0²): -22 (Ti-μ)² = 2πσ2 for - <μ<∞, 0 < o² < ∞ and i = 1, 2, ..., n. Prove that Σ₁=1 Xi Σ=1(xi - μ)2 μ = and 2 = n n
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