Question

Exercise 6.48. Let X1, X2, ..., Xin be independent exponential random variables, with parameter lį for Xi. Let Y be the minimum of these random variables. Show that Y ~ Exp(11 +...+ In).

Answer 1

Answer:

Given that:

Let
X1, 42, ...... Xin
be independent exponential random variables,with parameter $\lambda i$ for $\chi i$ . Let Y be the minimum of these random variables. Show that
Y ~ ExpX1 + ... + In)

Proof : The random variable Xi has cumulative distribution function

FXi(x) = P(Xi<) =1-e-lirir > 0

for i = 1, 2, . . . , n. Let the random variable
Y = min(X1, X2, ..., Xn)
. Then the cumulative distribution function of Y is
Fy(y) = P(Y <y)

= 1- P(Y > y)

= 1- P(min(X1, X2, ..., Xn) > y

= 1- P(X1 > y, X2 > y, ..., Xn >y)

=1 - P(X1>y) P(X2 > y...P(Xn > y

=1-e ye-124..e-Any

=1-e-(1.+....Any

pdf of y

fy(y) = d/dy F, (y)

fy(y) = d/dy(1-e-(41+..Anly)

f(y) = 0 - e-(1+..Any(-(41+ ... + In))

fy(y) = (41 +.... + n)e-(41+.. Any
,  $y\geq 0$

Which is pdf of  

exp( 11 + ....In)

Y = min(X1,...,xn) ~erp(11 + ....In)