Summer EC489.02 2011
Quiz 1 - Solution
You have 8 minutes to solve the following questions.
1. For a random variable X with E[X ] =
c; answer the following questions.
and V ar(X ) =
2
; and two constants b and
(a) (1.5 pts)E [aX + b] =?
Answer:
E [aX + b] = aE[X ] + b = a + b:
(b) (1.5 pts)V ar (aX + b) =?
Answer:
V ar(aX + b) = a2 V ar(X ) = a2
2
:
(c) (2 pts)E[aX + bX 2 ] =? (Hint: remember the denition of V ar(X ))
Answer: Since
V ar(X ) = E[X 2 ] fE[X ]g2 ;
we have
E[X 2 ] = V ar(X ) + fE[X ]g2 =
2
+
2
:
Then
E[aX + bX 2 ] = E[aX ] + E[bX 2 ]
= aE[X ] + bE[X 2 a ]
= + b( 2 + 2 ):
2. Let X be a random variable distributed with the Exponential distribution. Remember
that the moment generating function of the Exponential distribution is given by
MX (t) =
1
(1
t)
:
Use this information to nd
(a) (2.5 pts)E[X ],
Answer: Remember that
@MX (t)
@t
t=0
= E[X ]:
Then,
@
1
=
(1
t) 2 (
@t (1
t)
@MX (t)
= E[X ] = :
and
@t
t=0
1
)=
(1
t)2
(b) (2.5 pts)E[X 2 ]:
Answer: Remember that
@ 2 MX (t)
@t2
t=0
= E[X 2 ]:
Then,
1
@2
2 (1
@t
t)
=
=
and
@ 2 MX (t)
@t2
t=0
@
@t (1
2 (1
t)2
t) (
= E[X 2 ] = 2 2 :
2
3
)=
2
(1
2
t)3