Deduce the estimators for the paparmeters of normally distributed poulation using Maximum Liklihood Let X₁, ..., Xpthe pdf isx₁ be a random sample from a normal distribution N(µ,σ²)SOƒ (x | 14₂ 0²³) = √ √ √ 27° (x-µ)²1= exp{-20²with vector parametr (µ,0²)➤ The likelihood function isL(µ‚0²) = ƒ (x,,...,x„; µ‚0²) = Ï] [-i=1=In[ƒ(x₁, . . . , x₁; µ, σ²)]1συ2πn11(0√27) "exp(-20²2 Σ (x-μ)²}=)"i=1=2(x₁exp{-(x-μ)²}]20²In (2πσ²)120²Σ(x; -μ)²

Given Xi~N(μ, σ2 ); i=1,2,...,n

- Let X₁, he pdf is x₁ be a random sample from a normal distribution N(µ‚σ²) " 50 1 f (x | μ41₂0² ) = √ √ 27 exp{- with vector parametr (µ,σ²) The likelihood function is L(µ‚0²) = ƒ (Xx,,.., X,„;µ‚,0²) = Í] [ = exp{-- 3}] i=1 O√√2π = (x-μ)² 20² 1 (√√√27) exp{- συ2π 20² Σ (x²-44)²) i=1 In[ƒ(x; , . . . , x„; µ‚ σ²)] = − 12 | In (2πσ7) (xi-μ)²₂ 20² 1 20² Σ(x; – μ)

- Let X₁, he pdf is x₁ be a random sample from a normal distribution N(µ‚σ²) " 50 1 f (x | μ41₂0² ) = √ √ 27 exp{- with vector parametr (µ,σ²) The likelihood function is L(µ‚0²) = ƒ (Xx,,.., X,„;µ‚,0²) = Í] [ = exp{-- 3}] i=1 O√√2π = (x-μ)² 20² 1 (√√√27) exp{- συ2π 20² Σ (x²-44)²) i=1 In[ƒ(x; , . . . , x„; µ‚ σ²)] = − 12 | In (2πσ7) (xi-μ)²₂ 20² 1 20² Σ(x; – μ)

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

Section13.3: Special Probability Density Functions

Problem 30E

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Deduce the estimators for the paparmeters of normally distributed poulation using Maximum Liklihood

Let X₁, ..., Xp
the pdf is
x₁ be a random sample from a normal distribution N(µ,σ²)
SO
ƒ (x | 14₂ 0²³) = √ √ √ 27° (x-µ)²
1
= exp{-
20²
with vector parametr (µ,0²)
➤ The likelihood function is
L(µ‚0²) = ƒ (x,,...,x„; µ‚0²) = Ï] [-
i=1
=
In[ƒ(x₁, . . . , x₁; µ, σ²)]
1
συ2π
n
1
1
(0√27) "exp(-20²2 Σ (x-μ)²}
=)"
i=1
=
2
(x₁
exp{-(x-μ)²}]
20²
In (2πσ²)
1
20²
Σ(x; -μ)²

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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