STA2004F: Mother of All Workshop(MOAW) Solutions Fully Worked Out- Probability Theory (Melusi's section))

11
.
a .




Mx (t) ECeTX)
·




=




↑ (1 1)
g3x
-
=
-
1) -
+

xdo
-


y
S



=

fiedx
S

x
Fx(x) e id - e
fx(x)
- -



=
e =) = e .




x
- ex
-l -x
e
e




b


S
-
x
+ x)
:
Mx(+ ) - x - e
da
= b
& ·
e e



l+ u= e du = -e
as >( -) -



=-So
u> 0-




it i du as x+ -

- 1




-

go A
ite"du

r(1 2)
= -




(Probabiliz integral transform)
b: We know : Fx (X) -
U(0 1) ,




* >
-
U(p ,
= ) > NCo
-


,
= ] >
-
N(E,)
=
Fx(X) + I (U) 5(E ) + 1

·: (E (e- eX(j + 1 =

g(X)

, 12
.a

fa
.



y 1)
Fx(x) ds
-
-




=
e
1



=
e . (- e-S)))
)
=
= e( ex -
+ e

x+
1
1
+
-
=

-




( -

7)
1
-

= -
e



12 .
.
6 Y : = X [15 , - (a)

:
Fy(y) =
P(y = y)
=
P(y = y , X =5) +
p(yy X] ,



=
P(5 = X = y) [25 ,
0, (b) + P(b =
y X<5) It
, ,
?
a
)
if y<s


[Ex(5)
=

if a 325 since [Y 13 =
15X153
=
[x(53
Fx(y)
-



Ex(5) + Fx (5) if y=5




-
if y<


Lie
=


if D =y
< 5

1- 1)
if
-




y25y