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Formula Sheet Descriptive Measures 1 n Sample mean: x xi ni1 Sample variance: s 2
1 n 1i
n

( xi
1

x)2 cov( x, y ) 1 n 1i
n

Sample coefficient of variation: Sample coefficient ofcorrelation: Probability

cv
r

s Sample covariance: x cov( x, y ) s X sY

( xi
1

x )( y i

y)

P( A B) P( B) Bayes Law: If A1 , A2 ,, Ak are mutually exclusive and A1 A2  Ak is the wholesample space, then P( B Ai ) P( Ai ) P( Ai B) P( B A1 ) P( A1 )  P( B Ak ) P( Ak ) Random Variables
Conditional probability:

P( A B)

Expected Value: Covariance:

E( X )
all x

xp( x)
(x
XVariance:
)( y
Y

2

V (X )
all x

(x
X

) 2 p( x)
all x
Y

x 2 p ( x)

2

Cov( X , Y )
y x

) p ( x, y )
y x

x y p ( x, y )

Coefficient of correlation: Binomialdistribution: Poisson distribution:

Cov( X , Y )
X Y

P( X P( X

Exponential: density function Normal: density function
X

n! p x (1 p) n x , x x!(n x)! x e , x 0,1, x) x! f ( x) e x , x 0 x)f ( x) 1 2 e
1 x 2
2

0,1,, n

,

x

Sampling distributions Central Limit theorem: is distributed approximately standard normal for any population. to within W units:

n Size of a samplewhen the population standard deviation is known, to estimate

z n
For proportions:
ˆ p p
2

2

W

For two populations:

p (1 p ) n X1 X 2 (
2 1

is distributed approximately standardnormal

1 2 2

2

)

is distributed approximately standard normal

n1

n2

Conditions Normal population (or any population with n sufficiently large) with 2 known.

Confidence intervalH0

Test Statistic

Ha
0

Reject H 0 when
z0 z0
z0

z z
z
2

For
X z
2

:
0

z0

X n

0
0

n

0

For
X t S
2

:

0

t0
t0

t
t

n with n 1 degrees offreedom.

0

t0

X S n

0
0

Normal population with 2 unknown
(n 1) S
2 2 2

0

t0
2 0 2 0
2 0

t
2

2
2

2 0 2 0

2

For
2

:
(n 1) S
2 1 2 2

2

2

2 0...
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