4. Let (11, 12, ..,In) be independent samples from the population with distribution de- scribed by the density function f(x) = Be-®(z-B) 1>3 (a) Find the distribution of r; - nß. i=1 (b) Find the mean and variance of (c) Show that X(1) – B is exponentially distributed and clearly specify its parameter, X(1) is the minimum order statistic. (d) Hence write a function of Xq) - 3 that is distributed as the chi-square and specify its degrees of freedom.

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14. Let (71, 12, .., In) be independent samples from the population with distribution de-
scribed by the density function
f(x) = Be-0(x-8), r> B
(a) Find the distribution of r; - nß.
i=1
(b) Find the mean and variance of
i=1
(c) Show that X(1) – ß is exponentially distributed and clearly specify its parameter,
X(1) is the minimum order statistic.
(d) Hence write a function of X(1) - B that is distributed as the chi-square and specify
its degrees of freedom.
Transcribed Image Text:14. Let (71, 12, .., In) be independent samples from the population with distribution de- scribed by the density function f(x) = Be-0(x-8), r> B (a) Find the distribution of r; - nß. i=1 (b) Find the mean and variance of i=1 (c) Show that X(1) – ß is exponentially distributed and clearly specify its parameter, X(1) is the minimum order statistic. (d) Hence write a function of X(1) - B that is distributed as the chi-square and specify its degrees of freedom.
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