Question 1188982
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