The document describes various probability distributions that can arise from combining Bernoulli random variables. It shows how a binomial distribution emerges from summing Bernoulli random variables, and how Poisson, normal, chi-squared, exponential, gamma, and inverse gamma distributions can approximate the binomial as the number of Bernoulli trials increases. Code examples in R are provided to simulate sampling from these distributions and compare the simulated distributions to their theoretical probability density functions.

2. Twitter

9. 990
10
583.7
170.1
http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html
http://www.toeic.or.jp/toeic/about/data/data_avelist/data_dist01_04.html

10. 990
10
583.7
170.1
http://www.toeic.or.jp/toeic/about/data/data_avelist/data_ave01_04.html
http://www.toeic.or.jp/toeic/about/data/data_avelist/data_dist01_04.html

11. D = {x1, x2, · · · , xn}
¯x =
1
n
nX
i=1
xi
2
=
1
n
nX
i=1
(xi ¯x)2
=
v
u
u
t 1
n
nX
i=1
(xi ¯x)2

19. =
1
n
nX
i=1
|xi ¯x|

23. =
1
n
nX
i=1
(xi ¯x)2

24. p

25. =
v
u
u
t 1
N
NX
i=1
(xi ¯x)2
p

34. probability

35. ! 2 ⌦ = {!1, !2, · · · , !m}
⌦ = { , }
! 2 { , }
!(1)
= !(2)
=
!(n)
=

36. ⌦ = {1, 2, 3, 4, 5, 6}
!(1)
= !(2)
=
!(n)
=
⌦ = {!1, !2, · · · , !49870000}
!(1)
= !43890298 = 171cm
!(2)
= !29184638 = 168cm
!(n)
= !51398579 = 174cm

37. !(1)
= !(2)
=
!(n)
=!(3)
=
!1 !2 !3 !4 !5 !6 !7 !8 !9 !10
= {!1, !2, !3, · · · , !10}
! 2 ⌦ = {ID1, ID2, ID3, · · · , ID10}

38. ⌦ !
!

39. X = X(!)
⌦ !
!
X(!1) = 0
X(!2) = 0
X(!3) = 0
X(!4) = 0
X(!5) = 0
X(!6) = 0
X(!7) = 0
X(!8) = 0
X(!9) = 0
X(!10) = 100

40. !
{! 2 ⌦ : X(!) 2 A}
{X 2 A}
X(!) X

41. {! 2 ⌦ : X(!) 2 A}
!1 !2 !3 !4 !5 !6 !7 !8 !9 !10
A X(!) = 100Ac
X(!) = 0
!5 or !9

42. PX (A) = P(X 2 A) = P({! 2 ⌦ : X(!) 2 A})
⌦
!5, !9 !5, !9
PX (A) =
#({! 2 ⌦ : X(!) 2 A})
#( )
=
#(!5, !9)
#( )
=
2
10
= 0.2

43. PX(⌦) = 1
A1, A2, · · ·
PX ([iAi) =
X
i
PX (Ai)
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A12
0  PX(A)  1

45. X = X(!)
⌦
A
A
!1
!2
!3
!4
!5
!6
!7
!8
!11
!10
!9
!12
!13
!14
!15
!16
B
C
D
X(!) = 0
X(!) = 0
#A = #{! 2 ⌦ : X(!) = 0} = 7
#B = #{! 2 ⌦ : X(!) = 1} = 2
#C = #{! 2 ⌦ : X(!) = 2} = 4
#D = #{! 2 ⌦ : X(!) = 3} = 3

46. ⌦
A
A
!1
!2
!3
!4
!5
!6
!7
!8
!11
!10
!9
!12
!13
!14
!15
!16
B
C
DX(!) = 0
P(X = 0) = PX(A) =
#{! 2 ⌦ : X(!) = 0}
#⌦
=
7
16
P(X = 1) = PX (B) =
#{! 2 ⌦ : X(!) = 1}
#⌦
=
2
16
P(X = 2) = PX(C) =
#{! 2 ⌦ : X(!) = 2}
#⌦
=
4
16
P(X = 3) = PX(D) =
#{! 2 ⌦ : X(!) = 3}
#⌦
=
3
16

47. {x1, x2, · · · , xk}
P(X = xi) = f(xi)
F(x) = P(X  x)

48. P(x < X  x + x)
x + xx
x x ! 0
f(x) = lim
x!0
P(x < X  x + x)
x

49. x + xx
f(x)
F(x) = P(X  x) =
Z x
1
f(u)du
f(a < x < b) =
Z b
a
f(x)dx

50. http://www.math.wm.edu/~leemis/2008amstat.pdf

52. P(X = x) = px
(1 p)1 x
(x = 0, 1)

53. #
#
p = 0.7
trial_size = 10000
set.seed(71)
#
data <- rbern(trial_size, p)
#
dens <- data.frame(y=c((1-p),p)*trial_size, x=c(0, 1))
#
ggplot() +
layer(data=data.frame(x=data), mapping=aes(x=x), geom="bar",
stat="bin", bandwidth=0.1
) + layer(data=dens, mapping=aes(x=x, y=y), geom="bar",
stat="identity", width=0.05, fill="#777799", alpha=0.7)

55. (x = 0, 1, · · · , n)

57. #
p = 0.7
trial_size = 10000
sample_size = 30
set.seed(71)
#
gen_binom_var <- function() {
return(sum(rbern(sample_size, p)))
}
result <- rdply(trial_size, gen_binom_var())
#
dens <- data.frame(y=dbinom(seq(sample_size),
sample_size, 0.7))*trial_size
#
ggplot() +
layer(data=resuylt, mapping=aes(x=V1), geom="bar", stat = "bin",
binwidth=1, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=seq(sample_size)+.5, y=y),
geom="line", stat="identity", position="identity",colour="red"
) + ggtitle("Bernoulli to Binomial.")

60. P(X = x) =
e x
x!

62. trial_size = 5000; width <- 1;
#
p = 0.7; n = 10;
np <- p*n
# n!∞ p!0 np=
n = 100000; p <- np/n
#
gen_binom_var <- function() {
return(sum(rbern(n, p)))
}
result <- rdply(trial_size, gen_binom_var())
#
dens <- data.frame(y=dpois(seq(20), np))*trial_size
#
ggplot() +
layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=seq(20)+.5, y=y),
geom="line", stat="identity", position="identity",
colour="red"
) + ggtitle("Bernoulli to Poisson.")

64. f(x) =
1
p
2⇡ 2
exp
⇢
1
2
(x µ)2
2
( 1 < x < 1)

67. #
n <- 10000; p <- 0.7;
trial_size = 10000
width=10
#
gen_binom_var <- function() {
return(sum(rbern(n, p)))
}
result <- rdply(trial_size, gen_binom_var())
#
dens <- data.frame(y=dnorm(seq(6800,7200), mean=n*p,
sd=sqrt(n*p*(1-p)))*trial_size*width)
#
ggplot() +
layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=seq(6800,7200), y=y),
geom="line", stat="identity", position="identity",
colour="red") + ggtitle("Bernoulli to Normal.")

69. ( 1 < x < 1)
f(x) =
1
p
2⇡
exp
⇢
1
2
x2

71. #
n <- 10000; p <- 0.7
trial_size = 30000
width=0.18
#
gen_binom_var <- function() {
return(sum(rbern(n, p)))
}
result <- rdply(trial_size, gen_binom_var())
m <- mean(result$V1); sd <- sd(result$V1);
result <- (result - m)/sd
#
dens <- data.frame(y=dnorm(seq(-4,4,0.05), mean=0,
sd=1)*trial_size*width)
#
ggplot() +
layer(data=result, mapping=aes(x=V1), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=seq(-4,4,0.05), y=y),
geom="line", stat="identity", position=“identity",
colour="red"
) + ggtitle("Bernoulli to Standard Normal.")

73. f(x, k) =
(1/2)k/2
(k/2)
xk/2 1
e x/2
(0  x)
Xi
Z = X2
1 + · · · + X2
k

75. #
p <- 0.7; n <- 1000;
trial_size <- 100000; width <- 0.3;
df <- 3
# (3 )
gen_binom_var <- function() {
return(sum(rbern(n, p)))
}
gen_chisq_var <- function() {
result <- rdply(trial_size, gen_binom_var())
return(((result$V1 - mean(result$V1))/sd(result$V1))**2)
}
# df
result <- rlply(df, gen_chisq_var(),.progress = "text")
res <- data.frame(x=result[[1]] + result[[2]] + result[[3]])
# ( =3)
xx <- seq(0,20,0.1)
dens <- data.frame(y=dchisq(x=xx, df=df)*trial_size*width)
#
ggplot() + layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=xx, y=y),
geom="line", stat="identity", position="identity",
colour="blue" ) + ggtitle("Bernoulli to Chisquare")

78. f(x, ) =
⇢
e x
(x 0)
0 (x < 0)

80. trial_size = 7000; width <- .01;
#
p = 0.7; n = 10; np <- p*n;
# n!∞ p!0 np=
n = 10000; p <- np/n
#
gen_exp_var <- function() {
cnt <- 0
while (TRUE) {
cnt <- cnt + 1
if (rbern(1, p)==1){
return(cnt) # 1
}
}
}
data <- data.frame(x=rdply(trial_size, gen_exp_var())/n)
names(data) <- c("n", "x")
#
dens <- data.frame(y=dexp(seq(0, 1.5, 0.1), np)*trial_size*width)
ggplot() +
layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=seq(0, 1.5, 0.1), y=y),
geom="line", stat="identity", position="identity", colour="red"
) + ggtitle("Bernoulli to Exponential.")

82. f(x, ↵, ) =
↵
(↵)
x↵ 1
exp( x)
(0  x < 1)
↵X
i=1
Xi ⇠ (↵, )Xi ⇠ Exp( )

84. trial_size = 7000; width <- .035;
#
p = 0.7; n = 10; np <- p*n;
# n!∞ p!0 np=
n = 10000; p <- np/n; alpha <- 5
#
get_interval <- function(){
cnt <- 0
while (TRUE) {
cnt <- cnt + 1
if (rbern(1, p)==1){ return(cnt) }
}
}
gen_exp_var <- function() {
data <- data.frame(x=rdply(trial_size, get_interval())/n)
names(data) <- c("n", "x")
return(data)
}
result <- rlply(alpha, gen_exp_var())
data <- data.frame(x=result[[1]]$x + result[[2]]$x + result[[3]]$x + result[[4]]$x +
result[[5]]$x)
#
dens <- data.frame(y=dgamma(seq(0, 3,.01), shape=alpha, rate=np)*trial_size*width)
ggplot() +
layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=seq(0,3,.01), y=y),
geom="line", stat="identity", position="identity", colour="red"
) + ggtitle("Bernoulli to Gamma")

86. f(x, ↵, ) =
↵
(↵)
x (↵+1)
exp
✓
x
◆
(0  x < 1)
Xi ⇠ Exp( ) Z =
↵X
i=1
Xi ⇠ (↵, )
1/Z ⇠ IG(↵, )

88. trial_size = 7000; width <- .;
#
p = 0.7; n = 10; np <- p*n;
# n!∞ p!0 np=
n = 10000; p <- np/n; alpha <- 5
#
get_interval <- function(){
cnt <- 0
while (TRUE) {
cnt <- cnt + 1
if (rbern(1, p)==1){ return(cnt) }
}
}
gen_exp_var <- function() {
data <- data.frame(x=rdply(trial_size, get_interval())/n)
names(data) <- c("n", "x")
return(data)
}
result <- rlply(alpha, gen_exp_var())
data <- data.frame(x=1/(result[[1]]$x + result[[2]]$x + result[[3]]$x +
result[[4]]$x + result[[5]]$x))
#
dens <- data.frame(y=dinvgamma(seq(0, 23,.01), shape=5, rate=1/np)*trial_size*width)
ggplot() +
layer(data=data, mapping=aes(x=x), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=seq(0,3,.01), y=y),
geom="line", stat="identity", position="identity", colour="red"
) + ggtitle("Bernoulli to Inversegamma")

90. f(x) =
⇢
1 (0  x  1)
0 (otherwise)

91. Z = x1(1/2)1
+ x2(1/2)2
+ · · · + xq(1/2)q

92. width <- 0.02
p <- 0.5;
sample_size <- 1000
trial_size <- 100000
gen_unif_rand <- function() {
# sample_size 2
#
return (sum(rbern(sample_size, p) * (rep(1/2, sample_size)
** seq(sample_size))))
}
gen_rand <- function(){
return( rdply(trial_size, gen_unif_rand()) )
}
system.time(res <- gen_rand())
ggplot() +
layer(data=res, mapping=aes(x=V1), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + ggtitle("Bernoulli to Standard Uniform")

94. f(x, a, b) =
⇢
(b a) 1
(a  x  b)
0 (otherwise)

96. a <- 5
b <- 8;
width <- 0.05
p <- 0.5
sample_size <- 1000
trial_size <- 500000
gen_unif_rand <- function() {
# sample_size 2
#
return (sum(rbern(sample_size, p) * (rep(1/2, sample_size)
** seq(sample_size))))
}
gen_rand <- function(){
return( rdply(trial_size, gen_unif_rand()) )
}
system.time(res <- gen_rand())
res$V1 <- res$V1 * (b-a) + a
ggplot() +
layer(data=res, mapping=aes(x=V1), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + ggtitle("Bernoulli to Uniform") + xlim(4,9)

98. f(x, ↵, ) =
1
B(↵, )
x↵ 1
(1 x) 1
(0 < x < 1)
Xi ⇠ U(0, 1)iid
(i = 1, 2, · · · , ↵ + 1)

100. width <- 0.03; p <- 0.5
digits_length <- 30; set_size <- 3
trial_size <- 30000
gen_unif_rand <- function() {
# digits_length 2
#
return (sum(rbern(digits_length, p) *
(rep(1/2, digits_length) **
seq(digits_length))))
}
gen_rand <- function(){
return( rdply(set_size, gen_unif_rand())$V1 )
}
unif_dataset <- rlply(trial_size, gen_rand, .progress='text')
p <- ceiling(set_size * 0.5); q <- set_size - p + 1
get_nth_data <- function(a){ return(a[order(a)][p]) }
disp_data <- data.frame(lapply(unif_dataset, get_nth_data))
names(disp_data) <- seq(length(disp_data)); disp_data <- data.frame(t(disp_data))
names(disp_data) <- "V1"
x_range <- seq(0, 1, 0.001)
dens <- data.frame(y=dbeta(x_range, p, q)*trial_size*width)
ggplot() +
layer(data=disp_data, mapping=aes(x=V1), geom="bar", stat = "bin",
binwidth=width, fill="#6666ee", color="gray"
) + layer(data=dens, mapping=aes(x=x_range, y=y),
geom="line", stat="identity", position="identity", colour="red"
) + ggtitle("Bernoulli to Beta")

102. E[X] = X( )P( ) + X( )P( )
= 0 ⇥ 0.8 + 1, 000, 000 ⇥ 0.2
= 200, 000
E[X] =
X
x
xp(x)
µ

103. ✓
n
x
◆
=
n!
(n x)!x!
E[X] =
nX
x=0
xP(x) =
nX
x=0
x
✓
n
x
◆
px
(1 p)n x
=
nX
x=0
x
n!
(n x)!x!
px
(1 p)n x
=
nX
x=0
n
(n 1)!
(n x)!(x 1)!
px
(1 p)n x
= np
nX
x=0
✓
n 1
m 1
◆
p(x 1)
(1 p)(n 1) (x 1)
= np
= np
nX
x=1
✓
n 1
m 1
◆
p(x 1)
(1 p)(n 1) (x 1)
= np

104. Var[X] = E[(X E[X])2
]
=
X
x
(x E[x])2
P(x)
= 2
µ

105. Var[x] = E[(X E[X])2
]
=
Z 1
1
(x E[x])2
f(x)dx
= 2
E[X] =
Z 1
1
xf(x)dx
= µ

106. E[g(X)] =
Z 1
1
g(x)f(x)dx
g(X) = (X E[X])2
E[ · ] =
Z 1
1
· f(x)dx

107. g(x) = xk
E[g(X)] = E[Xk
] =
Z 1
1
xk
f(x)dx
µ0
k

108. g(x) = (x E[x])k
E[g(X)] = E[(X E[X]])k
] =
Z 1
1
(x E[x])k
f(x)dx
µk

109. E[cX] = cE[X]
* E[cX] =
Z 1
1
cxf(x)dx = c
Z 1
1
xf(x)dx
= cE[X]

110. Var[cX] = c2
Var[X]
* Var[cX] =
Z 1
1
(cx E[cx])2
f(x)dx
=
Z 1
1
(cx cµ)2
f(x)dx
=
Z 1
1
c2
(x µ)2
f(x)dx
= c2
Z 1
1
(x µ)2
f(x)dx
= c2
Var[X]

112. P(x < X 5 x + x, y < Y 5 y + y)
x, y ! 0
f(x, y) = lim
x, y!0
P(x < X 5 x + x, y < Y 5 y + y)
f(x, y)

113. g(x) =
Z 1
1
f(x, y)dy
h(y) =
Z 1
1
f(x, y)dx
g(x)
h(y)

114. EX,Y [ g(X, Y )] =
Z 1
1
Z 1
1
g(x, y)f(x, y)dxdy
g(x, y) = x0.8
y0.8 (x, y) ⇠ N((4, 4), S) S =

1 0.5
0.4 1
EX,Y [ g(X, Y )] = 8.02

115. g(X, Y ) = (X µX)(Y µY )
Cov[X, Y ] = E[(X µX)(Y µY )]

116. g(X, Y ) = (X µX)(Y µY )
µX µX
µX µX
µY
µY
µY
µY
S1 = S2 =
S3 = S4 =

1 0.8
0.8 1

1 0.8
0.8 1

1 0
0 1

1 0.999
0.999 1
Cov[X, Y ] = E[(X µX)(Y µY )]
(x, y) ⇠ N((4, 4), S)

117. f(x, y)
f(x, y) = g(x)h(y)

118. f(x, y) = g(x)h(y)
= 0

119. (x1, x2, · · · , xn)
x1
f(x1) =
Z
· · ·
Z
f(x1, · · · , xn)dx2 · · · dxn
x1
f(x1, · · · , xn) = f(x1) · · · f(xn)
x1 · · · xn

120. x1 · · · xn
g1(x1), · · · , gn(xn) x1 · · · xn
E[
nY
i=1
gi(xi)] =
nY
i=1
E[gi(xi)]
E[g1(x1)] E[gn(xn)]
E[
nY
i=1
gi(xi)] =
Z 1
1
· · ·
Z 1
1
g1(x1) · · · gn(xn)f(x1, · · · , xn)dx1 · · · dxn
=
Z 1
1
g1(x1)f(x1)dx1 · · ·
Z 1
1
gn(xn)f(xn)dxn
=
nY
i=1
E[gi(xi)]
f(x1) · · · f(xn)

121. x1 · · · xn
xi µi 2
i i = 1, 2, · · · , n
c = (c1, · · · , cn) c1x1 + · · · + cnxn
c1µ1 + · · · + cnµn
c2
1
2
1 + · · · + c2
n
2
n

122. E[c1x1 + · · · + cnxn]
=
Z 1
1
· · ·
Z 1
1
(c1x1 + · · · + cnxn)f(x1 · · · , xn)dx1 · · · dxn
= c1
Z 1
1
· · ·
Z 1
1
x1f(x1 · · · , xn)dx1 · · · dxn · · ·
cn
Z 1
1
· · ·
Z 1
1
xnf(x1 · · · , xn)dx1 · · · dxn
=c1
Z 1
1
x1dx1 · · · cn
Z 1
1
xndxn
=c1µ1 + · · · + cnµn
f(x1) · · · f(xn)
f(x1) · · · f(xn)
µ1 µn
=c1
Z 1
1
x1dx1 · · · cn
Z 1
1
xndxn
=c1µ1 + · · · + cnµn

123. Var[c1x1 + · · · + cnxn]
= E[{(c1x1 + · · · + cnxn) E[c1x1 + · · · + cnxn]}2
]
= E[{c1(x1 µ1) + · · · + c1(x1 µ1)}2
]
= E[
nX
i=1
c2
i (xi µi)2
+
X
i6=j
cicj(xi µj)(xi µj)]
=
nX
i=1
c2
i E[(xi µi)2
] +
X
i6=j
cicjE[(xi µj)(xi µj)]
= c2
1
2
1 + · · · + c2
n
2
n
c1µ1 + · · · + cnµn
= E[xi µi]E[xj µj] = 0= 2
i

125. x1 · · · xn
x1 · · · xn
xi
µ 2
(µ, 2
)

126. x1 · · · xn
T = x1 + · · · + xn
E[T] = E[x1 + · · · + xn]
= E[x1] + · · · + E[xn]
= nµ
Var[T] = Var[x1 + · · · + xn]
= Var[x1] + · · · + Var[xn]
= n 2
2
1 = · · · = 2
n
c1 = · · · = cn = 1
Var[c1x1 + · · · + cnxn]
= c2
1
2
1 + · · · + c2
n
2
n

127. ¯x =
1
n
nX
i=1
xi =
1
n
T
E[¯x] =
1
n
E[T] = n ·
1
n
µ = µ
Var[¯x] = Var[
1
n
T] =
1
n2
Var[T] =
2
n
µ
2

128. Var[¯x] =
2
n
=
0.0833
500
= 0.000166
E[¯x] = 0.5

129. Var[¯x]

131. µ 2
P(|x µ| > ) 5
1
2
µ 2
1/ 2
= 1 ) P(|x µ| > ) 5 1
= 2 ) P(|x µ| > ) 5 1/4
= 3 ) P(|x µ| > ) 5 1/9

132. 2
=
Z 1
1
(x µ)2
f(x)dx
=
Z
I1
(x µ)2
f(x)dx +
Z
I2
(x µ)2
f(x)dx +
Z
I3
(x µ)2
f(x)dx
2
=
Z
I1
(x µ)2
f(x)dx +
Z
I3
(x µ)2
f(x)dx
=
Z
I1
2 2
f(x)dx +
Z
I3
2 2
f(x)dx
= 2 2
[P(x 2 I1) + P(x 2 I3)]
I1 = ( 1, µ ),
I2 = [µ , µ + ],
I3 = (µ + , 1)
= P(|x µ| > )
P(|x µ| > ) 5
1
2
)

133. x1 · · · xn µ
2
" > 0
lim
n!1
P{|¯xn µ| = "} = 0
¯xn =
1
n
nX
i=1
xi
¯xn µ
¯xn ! µ in P

134. " > 0
P(|¯xn µ| > ")
= P(|¯xn µ| > "
p
n
p
n
)
5
2
"2n
= 2
¯x=
=
1
2

137. f(x) =
1
p
2⇡ 2
exp
✓
(x µ)2
2 2
◆
f(x) =
1
p
2⇡
exp
✓
x2
2
◆
1 < x < 1
1 < x < 1

138. f(y) = y2

139. f(x) = x2
f(y) = y2

140. f(y) = exp( y2
)

141. z =
p
2y
f(z) = exp
✓
1
2
z2
◆

142. Z 1
1
e y2
dy =
p
⇡
Z 1
1
exp
✓
z2
2
◆
dz =
p
2⇡
Z 1
1
1
p
2⇡
exp
✓
z2
2
◆
dz = 1
dz =
p
2dy

143. Z 1
1
1
p
2⇡
exp
✓
z2
2
◆
dz

144. z =
x µ dz
dx
=
1
f(x) =
Z 1
1
1
p
2⇡ 2
exp
✓
(x µ)2
2 2
◆
dx
1/

146. D = (x1, · · · , xn) µ 2
¯x µ
/
p
n
, n ! 1 N(0, 1)
= 0.1, µ =
1
= 10, 2
=
1
2
= 100 ¯x = p
n
=
r
1
2n
=
r
1
0.01 ⇥ 10000
=
r
1
100
=
1
10

147. g(x) = ext
E[ext
] =
Z 1
1
ext
f(x)dx
Mx(t) = E[ext
]
Mx(t)
My(t)
x
t = 0
y

148. g(x) = ext
ext
= 1 + xt +
t2
2!
x2
+ · · · +
tk
k!
xk
+ · · ·
Mx(t) = E[ext
]
= E[1 + xt +
t2
2!
x2
+ · · · +
tk
k!
xk
+ · · · ]
= 1 + tE[x] +
t2
2!
E[x2
] + · · · +
tk
k!
E[xk
] + · · ·
= 1 + xµ0
1 +
t2
2!
µ0
2 + · · · +
tk
k!
µ0
k + · · ·

149. Mx(t)
d
dtk
Mx(t) = E[xk
ext
]
t = 0
d
dtk
Mx(0) = E[xk
] = µ0
k

150. x ⇠ N(µ, )
Mx(t) = E[ext
] =
Z 1
1
ext 1
p
2⇡ 2
exp
✓
1
2
(x µ)2
2
◆
dx
z =
x µ
x = µ + z dx = dz

151. Mx(t) =
Z 1
1
e(µ+ z)t 1
p
2⇡ 2
exp
✓
1
2
z2
◆
dz
= eµt
Z 1
1
1
p
2⇡
exp
✓
tz
1
2
z2
◆
dz
= eµt
Z 1
1
1
p
2⇡
exp
✓
1
2
[z2
2 tz 2
t2
+ 2
t2
]
◆
dz
= eµt
Z 1
1
1
p
2⇡
e
2t2
2 exp
✓
1
2
(z t)2
◆
dz
= eµt
e
2t2
2
Z 1
1
1
p
2⇡
exp
✓
1
2
(z t)2
◆
dz
z t = w dz = dw
Mx(t) = eµt
e
2t2
2
Z 1
1
1
p
2⇡
exp
✓
w2
2
◆
dw = eµt+
2t2
2

152. (f · g)0
= f0
· g + f · g0
(f g)0
(x) = f0
(g(x))g0
(x)
M0
x(t) = (µ + 2
t)eµt+
2t2
2
M00
x (t) = (µ + 2
t)2
⇣
eµt+
2t2
2
⌘
+ 2
⇣
eµt+
2t2
2
⌘
=
⇣
eµt+
2t2
2
⌘
{(µ + 2
t)2
+ 2
}

153. Var[x] = E[x2
] (E[x])2
= (µ2
+ 2
) (µ)2
= 2
Var[x] = E[(x E[x])2
]
= E[x2
2E[x]x + E[x]2
)
= E[x2
] 2E[x]2
+ E[x]2
= E[x2
] E[x]2
t = 0
E[x] = M0
x(0) = (µ + 2
· 0)eµ·0+
2·02
2 = µ
E[x2
] = M00
x (0) =
⇣
eµ·0+
2·02
2
⌘
{(µ + 2
· 0)2
+ 2
} = µ2
+ 2

154. D = (x1, · · · , xn) µ 2
¯x µ
/
p
n
, n ! 1
N(0, 1)
T = x1 + · · · + xn
T nµ
p
n
2T0
=
T nµ
p
n
=
¯x µ
1/
p
n

155. Mx(t)
My(t)
x
t = 0
y
T T0
=
T nµ
p
n
N(0, 2
)

156. Mxi
(t) = 1 + µ0
1t + µ0
2
t2
2!
+ µ0
3
t3
3!
+ · · ·
Mxi µ(t) = 1 + µ1t + µ2
t2
2!
+ µ3
t3
3!
+ · · ·
= 1 + 0 + 2 t2
2!
+ µ3
t3
3!
+ · · ·

157. xi µ
p
n
xi µ
p
n
Mxi µ
p
n
(t) = E[e
xi µ
p
n
t
]
= 1 + 2 t2
2!n
+ µ3
t3
3!n3/2
+ · · · + µk
tk
k!nk/2
+ · · ·
= 1 +
2
t2
2n
+
n
2n
=
1
2n
n n ! 0 n ! 0
= 1 +
2
t2
+ n
2n

158. T0
=
x1 µ
p
n
+
x2 nµ
p
n
+ · · · +
xn µ
p
n
=
nX
i=1
xi µ
p
n
MT 0 (t) = MPn
i=1
⇣
xi µ
p
n
⌘(t) = E[e
Pn
i=1
⇣
xi µ
p
n
⌘
t
]
=
nY
i=0
E[e
⇣
xi µ
p
n
⌘
t
] =
✓
1 +
1
n
2
t2
+ n
2
◆n
er
⌘ lim
n!1
⇣
1 +
r
n
⌘n
r
r
= lim
n!1
⇣
1 +
r
n
⌘n

159. n ! 1
lim
n!1
MT 0 = lim
n!1
✓
1 +
1
n
2
t2
+ n
2
◆n
= e
2t2
2
lim
n!1
n = 0
N(0, 2
)
T0
=
T nµ
p
n
2

160. n = 100000
sample_size = 1000
rvs_list = []
m_list = []
for i in range(n):
unif_rvs = st.uniform.rvs(4.5, size=sample_size) # 5
beta_rvs = st.beta.rvs(a=3, b=3, size=sample_size) # 0.5 β
gamma_rvs = st.gamma.rvs(a=3, size=sample_size) # 3
chi2_rvs = st.chi2.rvs(df=5, size=sample_size) #
exp_rvs = st.expon.rvs(loc=0, size=sample_size) # 1
rvs = np.array([unif_rvs, beta_rvs, gamma_rvs, chi2_rvs, exp_rvs]).flatten()
m_list.append(np.mean(rvs))
rvs_list.append(rvs)

161. #
n = 10000
sample_size = 1000
rvs_list = []
m_list = []
m_unif = st.uniform.rvs(4, 2, size=sample_size)
m_beta_a = st.uniform.rvs(4, 2, size=sample_size)
m_beta_b = st.uniform.rvs(4, 2, size=sample_size)
m_gamma = rd.randint(2,5,size=sample_size)
m_chi2_df = rd.randint(3,6,size=sample_size)
m_exp = st.uniform.rvs(4, 2, size=sample_size)
def gen_random_state():
return int(dt.now().timestamp() * 10**6) - 1492914610000000 + rd.randint(0, 1000000)
def create_rvs(n):
#rd.seed = int(dt.now().timestamp() * 10**6) - 1492914610000000 + rd.randint(0, 1000000)
print("[START]")
for _ in range(n):
unif_rvs = [st.uniform.rvs(m, size=1, random_state=gen_random_state()) for m in
m_unif] # 5
beta_rvs = [st.beta.rvs(a=a, b=b, size=1, random_state=gen_random_state()) for a, b
in zip(m_beta_a, m_beta_b)]# 0.5 β
gamma_rvs = [st.gamma.rvs(a=a, size=1, random_state=gen_random_state()) for a in
m_gamma] # 3
chi2_rvs = [st.chi2.rvs(df=d, size=1, random_state=gen_random_state()) for d in
m_chi2_df] #
exp_rvs = [st.expon.rvs(loc=l, size=1, random_state=gen_random_state()) for l in
m_exp] # 1
rvs = np.array([unif_rvs, beta_rvs, gamma_rvs, chi2_rvs, exp_rvs]).flatten()
l_mean.append(np.mean(rvs))
l_rvs.append(rvs)
print("[END]")

162. n_jobs = 20
n_each = int(n/n_jobs)
jobs = [Process(target=create_rvs, args=(n_each,)) for _ in range(n_jobs)]
manager = Manager()
l_rvs = manager.list(range(len(jobs)))
l_mean = manager.list(range(len(jobs)))
start_time = time.time()
for j in jobs:
j.start()
time.sleep(0.2)
for j in jobs:
j.join()
finish_time = time.time()
print(finish_time - start_time)
m_list = l_mean[n_jobs:]
rvs_list = np.array(l_rvs[n_jobs:])
print(rvs_list.shape)

164. D = (x1, · · · , xn)

165. ✓0 = ˆ✓(X1, · · · , Xn)
ˆ✓lower(X1, · · · , Xn) 5 ✓0 5 ˆ✓upper(X1, · · · , Xn)

166. ˆ✓(X)

167. E[(ˆ✓(X) ✓)2
]

168. E[(ˆ✓(X) ✓)2
]
= E[{(E[ˆ✓(X)] ✓) + (ˆ✓(X) E[ˆ✓(X)])}2
]
= E[(E[ˆ✓(X)] ✓)2
+ 2(E[ˆ✓(X)] ✓)(ˆ✓(X) E[ˆ✓(X)]) + (ˆ✓(X) E[ˆ✓(X)])2
]
= (E[ˆ✓(X)] ✓)2
+ Var[ˆ✓(X)]
E[ˆ✓(X)] ✓
E[(ˆ✓(X) ✓)2
] = Var[ˆ✓(X)]

169. E[¯x] =
1
n
E[T] = n ·
1
n
µ = µ
¯x
s2
=
1
n 1
nX
i=1
(xi ¯x)2

171. lim
n!1
P{|¯xn µ| = "} = 0 ¯xn ! µ in P
ˆ✓n(X) n ! 1
ˆ✓n(X) ! ✓ in P
ˆ✓n(X)
¯xn µ

172. Var[ˆ✓(X)]
ˆ✓(X)

175. D = (x1, · · · , xn) xi
f(xi)
nY
i=1
f(xi)
nY
i=1
f(xi|✓)
xi
`(✓|x1, x2, · · · , xn) =
nY
i=1
f(xi|✓)

176. x1, x2, · · · , x10
f(x1, x2, · · · , x10|µ, 2
) =
10Y
i=1
1
p
2⇡ 2
exp
✓
1
2
(xi µ)2
2
◆

177. `(µ, 2
|x1, x2, · · · , x10) =
10Y
i=1
1
p
2⇡ 2
exp
✓
1
2
(xi µ)2
2
◆

179. ✓⇤
= arg max
✓
`(✓|x1, x2, · · · , xn)
log `(✓|x1, · · · , xn) ⌘ L(✓|x1, · · · , xn)
`

181. µ, 2
L(µ, 2
|x1, x2, · · · , x10) =
n
2
(2⇡)
n
2
log 2 1
2 2
nX
i=1
(xi µ)2
@L
@µ
=
1
2 2
nX
i=1
(xi µ)2
)
nX
i=1
xi = nµ
) µ⇤
=
1
n
nX
i=1
xi
`(µ, 2
|x1, x2, · · · , xn) =
nY
i=1
1
p
2⇡ 2
exp
✓
1
2
(xi µ)2
2
◆

182. @L
@ 2
=
n
2
1
2
+
1
2( 2)2
nX
i=1
(xi µ)2
= 0
)
1
2( 2)2
nX
i=1
(xi µ)2
=
n
2 2
) 2⇤
=
1
n
nX
i=1
(xi µ)2
2⇤

184. D = (x1, · · · , xn)µ 2
µ

185. u ⇠ N(0, 1)
t =
u
p
v/m
v ⇠ 2
(m)
f(t) =
m+1
2
p
m⇡ m
2
✓
t2
m
+ 1
◆ m+1
2

186. u ⇠ N(0, 1) v ⇠ 2
(m) v > 01 < u < +1
f(u, v) =
1
p
2⇡
exp
✓
u2
2
◆
(1/2)n/2
(n/2)
vn/2 1
e v/2
t =
u
p
v/m
x = v
f(t) =
m+1
2
p
m⇡ m
2
✓
t2
m
+ 1
◆ m+1
2
(z) =
Z 1
0
tz 1
e t
dt

187. µ
D = (x1, · · · , xn) xi ⇠ N(µ, 2
)
¯x ⇠ N(µ, 2
/n)¯x
1
2
nX
i=1
(xi ¯x)2
⇠ 2
n 1

188. u =
¯x µ
/
p
n
⇠ N(0, 1) v =
1
2
nX
i=1
(xi ¯x)2
⇠ 2
n 1
t =
u
p
v/(n 1)
=
¯x µ
/
p
n
·
"
1
2
1
(n 1)
nX
i=1
(xi ¯x)2
# 1/2
=
¯x µ
1/
p
n
·
1
p
s2
=
¯x µ
s/
p
n
⇠ tn 1
s2
=
1
n 1
nX
i=1
(xi ¯x)2
s2

189. P
✓
tn 1;↵/2 5
¯x µ
s/
p
n
5 tn 1;↵/2
◆
= 1 ↵
tn 1;↵/2 tn 1;↵/2
↵/2 ↵/2
1 ↵
1 ↵
1 ↵
P
✓
¯x tn 1;↵/2
s
p
n
5 µ 5 ¯x + tn 1;↵/2
s
p
n
◆
= 1 ↵
[ tn 1;↵/2, tn 1;↵/2]
µ
1 ↵

190. P
✓
tn 1;↵/2 5
¯x µ
s/
p
n
5 tn 1;↵/2
◆
= 1 ↵
tn 1;↵/2 tn 1;↵/2
↵/2 ↵/2
1 ↵
1 ↵
1 ↵
P
✓
¯x tn 1;↵/2
s
p
n
5 µ 5 ¯x + tn 1;↵/2
s
p
n
◆
= 1 ↵
[ tn 1;↵/2, tn 1;↵/2]
µ
1 ↵

193. = 1 µ = 0
H0 : µ0 = 0
H1 : µ 6= µ0

194. ¯x = /
p
n
/
p
10 ; /3.16

195. ↵/2 ↵/2
H0 : µ0 = 0

202. H1 : µ = 1

203. H1 : µ = 0.5

204. H1 : µ = 3
µ0H1 : µ = 3
H0 : µ0 = 0

206. e↵ect size : =
µ µ0

212. …
…
…
…
…
…
…
…

221. r =
1
n
Pn
i=1(xi ¯x)(yi ¯y)
q
1
n
Pn
i=1(xi ¯x)2
q
1
n
Pn
i=1(yi ¯y)2

222. r =
1
n
Pn
i=1(xi ¯x)(yi ¯y)
q
1
n
Pn
i=1(xi ¯x)2
q
1
n
Pn
i=1(yi ¯y)2

223. r =
1
n
Pn
i=1(xi ¯x)(yi ¯y)
q
1
n
Pn
i=1(xi ¯x)2
q
1
n
Pn
i=1(yi ¯y)2

224. r =
1
n
Pn
i=1(xi ¯x)(yi ¯y)
q
1
n
Pn
i=1(xi ¯x)2
q
1
n
Pn
i=1(yi ¯y)2

225. 1
n
nX
i=1
(xi ¯x)(yi ¯y)

226. 1
n
nX
i=1
(xi ¯x)(yi ¯y)

227. 1
n
nX
i=1
(xi ¯x)(yi ¯y)

228. 1
n
nX
i=1
(xi ¯x)(yi ¯y)

229. 1
n
nX
i=1
(xi ¯x)(yi ¯y)

230. 1
n
nX
i=1
(xi ¯x)(yi ¯y)

231. 1
n
nX
i=1
(xi ¯x)(yi ¯y)