Math Problem Solver

Let

𝑋

1

X

1

​

and

𝑋

2

X

2

​

be independent continuous random variables and each has a density of the form

𝑓

(

𝑥

)

=

{

𝜆

𝑒

−

𝜆

𝑥

,

0

<

𝑥

<

∞

0

,

otherwise

f(x)={

λe

−λx

,

0,

​

0<x<∞

otherwise

​

(a) Find the joint p.d.f of

𝑋

1

X

1

​

and

𝑋

2

X

2

​

. [2 Marks]

(b) Find the joint p.d.f. of two new random variables

𝑌

1

=

𝑋

1

𝑋

2

,

𝑌

2

=

𝑋

1

+

𝑋

2

.

Y

1

​

=

X

2

​

X

1

​

​

,Y

2

​

=X

1

​

+X

2

​

.

[4 Marks]

(c) Hence determine the marginal densities of

𝑌

1

Y

1

​

and

𝑌

2

Y

2