... =
K
N
K
2
−N
2
−N
N
N /2
N /2
r=−N /2
e
−r
2
/2
2
2
−N
N
N /2
√
2 σ, (1.41)
where σ =
N/4, from which equation (1.40) follows. The distinction between
N /2 and N /2 is not important ... links.
34 2 — Probability, Entropy, and Inference
revealing a binary variable whose probability distribution is {
1
/
2,
1
/
2} . This
revelation has an entropy...
... =
−xα
−∞
dy
1
√
2 σ
2
e
−
y
2
2σ
2
= Φ
−
xα
σ
. (9.53)
Random coding
Solution to exercise 9 .20 (p.156). The probability that S = 24 people whose
birthdays are drawn at random from A = 365 ... |x) is the same for each value of x: H(Y |x =0) is H
2
(0.15),
and H(Y |x =1) is H
2
(0.15). So
I(X; Y ) = H(Y ) −H(Y |X)
= H
2
(0 .22 ) − H
2
(0.15)
= 0.76 − 0.61 = 0.15 bits. (...
... equilibrium [σ
2
(t + 1) = σ
2
(t)]
then the factor in (19 .2) is
α(1 + β /2)
1 /2
/
√
2 =
2
(π + 2)
0. 62.
Defining this constant to be η ≡ 2/ (π + 2) , we conclude that, under sex and natural
selection, ... below
1/β
1 /2
to above 1/β
1 /2
. The data
variance is indicated by the
ellipse.
r
1
(x) =
exp(−β(x
1
− m)
2
/2)
exp(−β(x
1
− m)
2
/2) + exp(−β(x
1
+ m)
2
/2)...
... thus
computes
P (r ≤ 3 |n = 12, H
0
) =
3
r=0
n
r
1
/
2
n
=
12
0
+
12
1
+
12
2
+
12
3
1
/
2
12
= 0.07, (37 .27 )
and reports ‘at the significance level of 5%, there is not ... θ
could either be 29 or 28 , and both possibilities are equally likely (if the prior
probabilities of 28 and 29 were equal). The posterior probability of θ is 50%
on 29 an...