(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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(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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