Question 1188982: (a) Let Y1 < Y2 < Y3 be the order statistics of a sample of a fixed size n = 3 from a
distribution with the pdf
f(x) = 1; 0 < x < 1
0; Otherwise.
Obtain the following:
(a) Define the range
(b) Find the pdf of the range
(c) Find the cdf of the range
(d) Find the hazard function of the sample range
(e) Find the reverse hazard function of the sample
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! Here's how to find the range, its pdf, cdf, hazard function, and reverse hazard function for the given distribution and sample size:
**(a) Range:**
The range (R) of a sample is the difference between the largest and smallest values. In this case, R = Y₃ - Y₁.
**(b) PDF of the Range:**
The probability density function (pdf) of the range R is given by:
g(r) = n(n-1) ∫₀^(1-r) [F(x+r) - F(x)]^(n-2) * f(x+r) * f(x) dx
Where:
* n = sample size = 3
* F(x) = CDF of the original distribution = x for 0 < x < 1, 0 otherwise.
* f(x) = PDF of the original distribution = 1 for 0 < x < 1, 0 otherwise.
Substituting:
g(r) = 3(2) ∫₀^(1-r) [(x+r) - x]^(3-2) * 1 * 1 dx
g(r) = 6 ∫₀^(1-r) r dx
g(r) = 6r [x]₀^(1-r)
g(r) = 6r(1-r) for 0 < r < 1
g(r) = 0 otherwise
**(c) CDF of the Range:**
The cumulative distribution function (CDF) of the range R is given by:
G(r) = ∫₀ʳ g(x) dx
G(r) = ∫₀ʳ 6x(1-x) dx
G(r) = 6 ∫₀ʳ (x - x²) dx
G(r) = 6 [x²/2 - x³/3]₀ʳ
G(r) = 6(r²/2 - r³/3)
G(r) = 3r² - 2r³ for 0 < r < 1
G(r) = 0 for r ≤ 0
G(r) = 1 for r ≥ 1
**(d) Hazard Function of the Sample Range:**
The hazard function h(r) is defined as:
h(r) = g(r) / (1 - G(r))
h(r) = 6r(1-r) / (1 - (3r² - 2r³))
h(r) = 6r(1-r) / (1 - 3r² + 2r³) for 0 < r < 1
h(r) = 0 otherwise
**(e) Reverse Hazard Function of the Sample Range:**
The reverse hazard function rh(r) is defined as:
rh(r) = g(r) / G(r)
rh(r) = 6r(1-r) / (3r² - 2r³)
rh(r) = 6(1-r) / (3r - 2r²)
rh(r) = 6(1-r) / r(3 - 2r) for 0 < r < 1
rh(r) = 0 otherwise
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