▷For the pdf fy (y; 6) = 241-0²050 ≤ y ≤12DISJI-7² a consistent estimator?Is1-Show all work.

Given function: fy=2y1-θ2, θ≤y≤1 Consider, Ey=∫θ1y*fydy =∫θ1y*2y1-θ2dy =2θ2+θ+13θ+1…

Answered: For the pdf fy (y; 6) = 24 2 1-0 ²…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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▷
For the pdf fy (y; 6) = 24
1-0²
05
0 ≤ y ≤1
2
DISJI-7² a consistent estimator?
Is
1-
Show all work.

Expert Solution

Step 1

Given function:

$f (y) = \frac{2 y}{1 - θ^{2}}, θ \leq y \leq 1$

Consider,

$E (y) = \int_{θ}^{1} y * f (y) d y = \int_{θ}^{1} y * \frac{2 y}{1 - θ^{2}} d y = \frac{2 (θ^{2} + θ + 1)}{3 (θ + 1)}$

$E (y^{2}) = \int_{θ}^{1} y^{2} * f (y) d y = \int_{θ}^{1} y^{2} * \frac{2 y}{1 - θ^{2}} d y = \frac{(θ^{2} + 1)}{2}$

$V (y) = E (y^{2}) - {[E (y)]}^{2} = \frac{θ^{4} + 2 θ^{3} - 6 θ^{2} + 2 θ + 1}{18 {(θ + 1)}^{2}}$

Condition for consistent estimator:

${\hat{θ}}_{n}$ is said to be consistent estimator for $θ$ then ${\hat{θ}}_{n} \overset{P}{\to} θ$ .

$\underset{n \to \infty}{l i m} P (|{\hat{θ}}_{n} - θ| > ε) = 0$ .

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