Question

2.2.8. Suppose X1 and X2 have the joint pdf

f(x1, x2) = "

e−x1 e−x2

x1 > 0, x2

> 0

0 elsewhere

.

For constants w1 > 0 and w2 > 0, let W = w1X1 + w2X2.

(a) Show that the pdf of W is

fW (w) = "

1

w1−

w2

(e−w/w1 − e−w/w2) w > 0

0 elsewhere

.

(b) Verify that fW (w) > 0 for w > 0.

(c) Note that the pdf fW (w) has an indeterminate form when w1 = w2. Rewrite

fW (w) using h defifined as w1

−w2 = h

. Then use l’Hˆ

opital’s rule to show that

when w1 = w2, the pdf is given by

f

W

(w)=(w/w

2

1

) exp{−w/w1} for w > 0

and zero elsewhere.

Answer 1

Answer:

Given That,

Suppose X₁ and X₂ have the joint pdf

For constants W₁>0 and W₂ >0,

let W= W₁ X₁ + W₂ X₂

(a).

Show that the pdf of W is

Jacobian matrix given by,

where o<u<w<

So, we get the marginal pdf of w from the joint pdf of w and u.

where 0<w<

(b).

Verify that f_W(w)>0 for w>0.

When,

So,f_W(w)>0

(c).

Defined h=w₁-w₂

Hence, using I'Hopital's rule