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yes find 95% CI
ga
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1 yes
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Probability of
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Emax Z x= 0 198 % Le 1 64
. =
Confidence
=
.
Interval
· fewest 595 %
(2:
a = 0 .
Za = 1 .
96
n
samples =
=
assume : Stadev = # = 6
a = 0 .
298 % [e = 2 32
[i-m +]
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Fxy (0 y) Fy(y)
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=
,
62 I =
PDFFX is
S
sample mean
sta
Student -
E
Exi
I -En [m]
in :
marginal
· xy(X
m +
Y) dxdy 1
=
for samp mean Sample
,
, var.
= =? chi-s (20)
unknown var.
assume
quare E(Xi i)
mean ,
·
P(X , X * Xa Y, <
Yaye) xy(x y)dydx -
:? jc
,
[#
,
=
mean
n - 1
(x(x) = xydyfy() =xy(x +)dx
62 ?
· =
Enter list to stat
for samp va
,
2nd -
> Stat /list) >
-
m
aussian wf Sta =# = G
for pof
(3)
g m = mean
Expression
4
[m -
(E) () ,
m +
(2)()] = variance (8)
true mean : 10 true var =
approx
I -
(x-10/(2)(0) gauss [2 -(0)(E) , (0) (E)] +
f(x) (10)
e n = 100
=
My
I
Bivariate over
region Sample Mean Marginal Dist
unbiased-EE23 my
for
=
pression CDF
joint
consistent var(z) = -
as n >c
2
o
mean Mx = E(x] = ( Xfx(x)dx
Var(x) E(X] m
-
whena trying to est.
-
=
value true value its
expected = .
# 4) ?) (it) when-2(x12 & Y72 height : Frea Th : as sample
to the true value
size? estimators value
converges
[[X] = / *fx(x)dx
4(x + 2)(b)
·
captures some vol .
always but amount
changes... joint
stat. Indep Check
.
F(x,y) fx(x fy(y)
= 0 "
X- (lower bound) = X -
(-2) dist.
( ?)(in) when-2(x2 marg
Et fxt * fl -1 not indep.
.
# & -2 <> 2
fx(x) &P(x y = y) i
= = X
, ,
(x (2))(y (2))(it) +n(x+ 2)(y+2)+ +(xy+ 2x + 2y 2) +
fy())
- -
# 4) ?)(in) when x>2z-2 < x12
f() valued 3
=
+ 2 + 1 = 3+
fy(0) 36
fx (1) valuef 6 + 5 +4 2+5
= =
4) (y -
(2))(in) = +(y+ 2)
mean variance
Exp joint PDF 3(x) =
1(6) + 1(15) [[x] = P(n) + 1(15) var(x) 3(x2) E(x))
height
- = -
.
-2
.
PDF : ()π() E(y) 1(9) + 0(7)[[y2] 1(9) + 07) var(y) Exyz) E(y)2
3
= = =
cut X
-
cross
Covariance Cxy
of space = (*)
visual vol .
crosscuty the same
=) Cov(x y) =
E(xy) E(x)E(y) -
uncorrelated kindep. Cxy0
,
(x() coeff value of
take deriv .
of CDF Correlations Rxy uncorrelated =
(xy(x y) = ,
E(xy) =
( 1)(1)(1) + ( 1)(0)(2) + (1)( 1)(3) -
...
paffx(x) Coeff
-
marginal , car(x, y)
fx(x) =
(afxy(x y)dy , f) dy
=
bounds
of X
#
y joint gauss . RV paf
[Ix-mxM2Pym
it
_
marginal paf fy (y) unity =1- 1 bh = h =
↓
cry 2
=>(*) - fxy(x,y) 2xe =
fy(y) jfxy(x for x
y)d
=
same
,
bounds Y Convolution Bernoulli 500 people vote 260
yes find 95% CI
ga
P. ,
I
0-NO
find
1 yes
RVz X+ Y
-
210 p
Probability of
0 4
0)
1
P(0-X1
- = =
=y =
.
(260(1)
+
240(07] 52
g
, + 0
in [260(1
=
P(H)
.
H = 1 = 0 4
P(0(X : ) ThArea off)
- 1zy = 0) =
Pa 9
52)2)
.
*
-Po
, 5 = 0 P(T) = 0 C . - 5232 + 24010 - = 0 250 .
= )
(EP P(220 dy =
joint = PRE Fin CI =
[m-m +]
