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3.1
a) This is an instance of the Bernoulli experiment.
73
7
10
1
1
120
0.117
2
2
1024
C
⎛⎞⎛⎞
==
⎜⎟⎜⎟
⎝⎠⎝⎠
b)
P(both tosses are tails  one of the tosses in a tail) = P(both tosses are tails)/P(one of the tosses
in a tail) = 1/3
c) () 0
.
4
,() 0
.
2
5
PA
PB
max
(
)
0.25
PA B
∩=
min
(
)
0
∩
=
max
(
)
0.65
∪=
min
(
)
0.4
d) Probability density function
2
2
1(
)
()
e
x
p
2
2
x
fx
µ
σ
σπ
⎛⎞
−
=−
⎜⎟
⎝⎠
mean is
, variance is
2
.
3.2
a)
112314 1
1
T
ab
=×+×+× =
b) The dot product of two vectors
u
and
x
is defined as
cos
T
ux u x
θ
=
where
u
,
x
are the length of vector
u
and
x
,
and
is the measure of the angle between vectors
u
and
x
.
when
1, cos
1
u
,
max(
)
cos
T
ux
u x
x
when
1
u
−
,
min(
)
cos
T
x
=
0
u
,
min(
) 
cos
 0
T
=
=
c)
2
(1
)
x
x
df x
e
dx
e
+
b)
2
(, ) 1
2
xy
fxy
x
ye
yy
∂
=+ ⋅
∂
c)
cos(
) ( 1)
sin(
)
cos(
)
sin(
)
ff
x
f
y
yu
v
x
u
v
vx
vy
v
v
x
u
v
∂∂
∂
=⋅+⋅=
⋅
−
⋅
−
−
⋅
+
∂
− −
+
,
then plug in x and y.
d) Suppose
2
x
x
f
xa
eb
e
−
=+
2
2
x
x
df
ae
be
dx
−
, let
df
dx
be zero, we have
2
20
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 Spring '09
 yangliu
 Machine Learning

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