9th EditionAllan G. Bluman2,814 explanations

12th EditionMario F. Triola2,593 explanations

10th EditionAllan G. Bluman2,364 explanations

12th EditionCharles Henry Brase3,169 explanations

PROBABILITYLet W be a gamma random variable with parameters
$$
( t , \beta )
$$
, and suppose that conditional on
$$
W = w , X _ { 1 } , X _ { 2 } , \ldots , X _ { n }
$$
are independent exponential random variables with rate w. Show that the conditional distribution of W given that
$$
X_1=x _ { 1 } , X _ { 2 } = x _ { 2 } , \ldots , X _ { n } = x _ { n }
$$
is gamma with parameters
$$
\left( t + n , \beta + \sum _ { i = 1 } ^ { n } x _ { i } \right)
$$
.