Question 20 - Week 3 Practice | Prasnya
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Week 3
Question 20
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Est. 2 min
Marks:
+3.00
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Consider
n
n
bits
X
1
,
…
,
X
n
X_1,\ldots,X_n
, where each bit is equally likely to be
0
0
or
1
1
, and is independent of all other bits. Define
n
−
1
n-1
bits
Y
i
=
X
i
X
i
+
1
Y_i=X_iX_{i+1}
,
i
=
1
,
…
,
n
−
1
i=1,\ldots,n-1
. Let
N
X
N_X
and
N
Y
N_Y
be, respectively, the number of
1
1
s in
{
X
1
,
…
,
X
n
}
\{X_1,\ldots,X_n\}
and
{
Y
1
,
…
,
Y
n
−
1
}
\{Y_1,\ldots,Y_{n-1}\}
. Assuming
n
=
100
n=100
, what is the expected value of
N
X
N_X
and
N
Y
N_Y
?
Single correct
Medium Difficulty
07 July 2024
A
E
(
N
X
)
=
50
,
E
(
N
Y
)
=
49.5
E(N_X)=50,\ E(N_Y)=49.5
B
E
(
N
X
)
=
50
,
E
(
N
Y
)
=
24.75
E(N_X)=50,\ E(N_Y)=24.75
C
E
(
N
X
)
=
100
,
E
(
N
Y
)
=
50
E(N_X)=100,\ E(N_Y)=50
D
E
(
N
X
)
=
100
,
E
(
N
Y
)
=
99
E(N_X)=100,\ E(N_Y)=99
