find the MLE of o (is in the image) Problem 1Let X1, X2, ... , Xn be a random sample with pdff(x) = (0+1)xº-1,0sx<1,0 < 0 < 0%3D Find the MLE of 0.

Given information: f(x)=θ+1xθ-1, 0≤x≤1, 0<θ<∞ Consider, the likelihood function, Lx=∏i=1nfx…

Answered: Let X1, X2, ... , Xn be a random sample…

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find the MLE of o (is in the image)

Problem 1
Let X1, X2, ... , Xn be a random sample with pdf
f(x) = (0+1)xº-1,
0sx<1,
0 < 0 < 0
%3D

Expert Solution

Step 1

Given information:

$f (x) = (θ + 1) x^{θ - 1}, 0 \leq x \leq 1, 0 < θ < \infty$

Consider, the likelihood function,

$L (x) = \prod_{i = 1}^{n} f (x) = \prod_{i = 1}^{n} (θ + 1) x^{θ - 1} = {(θ + 1)}^{n} \prod_{i = 1}^{n} x^{θ - 1}$

Apply ln on both sides,

$\ln L (x) = \ln [{(θ + 1)}^{n} \prod_{i = 1}^{n} x^{θ - 1}] = n \ln (θ + 1) + \sum_{i = 1}^{n} \ln (x^{θ - 1})$

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Let X1, X2, ... , Xn be a random sample with pdf f(x) = (0+1)xº-1, 0sx<1,