Best solutions to - "Consider the probability density function f(x)=c(1+θx),−1≤x≤1 a. Find..."

a2linetagadaW

Answered question

2021-08-16

Consider the probability density function

f (x) = c (1 + θ x), - 1 \leq x \leq 1

a. Find the value of the constant c b. Find both the moment and maximum likelihood estimator for eslimators for

θ

2k1enyvp

Skilled2021-08-17Added 94 answers

Given:

f (x) = C (1 + θ x) - 1 \leq x \leq 1

a)A valid PDF satisfies

\int_{x} f (x) d x = 1

\int_{- 1}^{1} c (1 + θ x) d x = 1

C {[x + θ \frac{x^{2}}{2}]}_{- 1}^{1} = 1

c = \frac{1}{2}

b) Moment estimator for

θ, μ_{1}^{1} = \int_{- 1}^{1} x f (x) d x

μ_{1}^{1} = s \frac{1}{2} \int_{1}^{- 1} (x + θ x^{2}) d x = \frac{1}{2} {[\frac{x^{2}}{2} + θ \frac{x^{3}}{3}]}_{- 1}^{1}

μ_{1}^{1} = \frac{1}{2} [1 - (1) + \frac{θ}{3} (1 - (- 1))] = \frac{θ}{3}

Maximum likelihood estimator for

θ

f (x) = \frac{(1 + θ x)}{2}

L (θ) = \prod_{1 = 1}^{n} f (x_{1}) = \frac{(1 + θ x_{1}) (1 + θ x_{2}) (1 + θ_{3}) \dots \cdot (1 + θ x_{n})}{2}

L (θ) = \prod_{1 = 1}^{n} (1 + θ x_{1})

Take log, log

L (θ) = - n \log 2 + \sum_{i = 1}^{n} (x_{i}) \frac{1}{(1 + θ x_{i}) = 0}

Now

\frac{δ}{δ θ} L (x, θ) = 0 + \sum_{i = 1}^{n} (x_{i}) \frac{1}{(1 + θ x_{i})} = 0

Given

μ, \sum_{i = 1}^{n}

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