IEEM 225
Tutorial 3
2/5/16
1. Let the probability density of X be given by
f ( x)  c(4 x  2 x 2 ), 0<x<2
0
otherwise
(a) What is the value of c
(b) P{1/2<X<3/2}=?
2. Let c be a constant. Show that
(i) Var(cX)=C2Var(X)
(ii) Var(c+X)=Var(X)
3. A coin, having probability p of landing heads, is flipped until head appears for the
rth time. Let N denote the number of flips required. Calculate E[N]
4. Cacluate the moment generating function of the uniform distribution on (0,1).
Obtain E[X] and Var[X] by differentiating.
Solution:
1.
2
c  (4 x  2 x 2 )dx  1
0
2
c(2 x 2  x 3 ) |02  1
3
8
c 1
3
c=3/8
P{1/2<X<3/2}=
3 3/ 2
11
(4 x  2 x 2 )dx 

8 1/ 2
16
2.
Var(cX)=E[(cX-E[cX])2]=E[c2(X-E(X))2]=c2Var(X)
Var(c+X)=E[(c+X-E[c+X])2]=E[(X-E(X))2]=Var(X)
3.
N=

r
i 1
X i where Xi is the number of flips between the ( i  1)st and ith head.
Hence, Xi is geometric with mean 1/p. Thus,
r
r
E[N]= i 1 E[ X i ] 
p
4.
1
et  1
E[e tX ]   e tX dx 
0
t
IEEM 225
Tutorial 3
d
tet  e t  1
E[e tX ] 
dt
t2
d2
[t 2 (tet  e t  1)  2t (tet  e t  1)] t 2 e t  2(tet  e t  1)
tX
E
[
e
]


.
dt 2
t4
t3
To evaluate at t=0,
tet  e t  e t
et 1
E[X]= lim
 lim 
t 0
t 0 2
2
2t
t
2 t
t
t
2te  t e  2te  2e  2e t
et 1
E[X2]= lim

lim

t 0
t 0 3
3
3t 2
2
Hence, Var(X)=1/3-(1/2) =1/12
2/5/16