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Formula Sheet
December 8, 2011
Abstract
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1
Descriptive Statistics
1.1
Central Tendency
1. (Arithmetic) Mean:
PN
xi
Population: = i=1
Pn N
xi
Sample: x = i=1
n
Pn
wi xi
2. Weighted mean:xw = Pi=1
n
i=1 wi
3. Median:
The 0:5 (n + 1)th data.
If n is even, then we take 0:5 (n + 1)th data directly
If n is odd, then we take mean of 0:5nth and (0:5n) + 1th data
4. Mode:
The most frequent data. (However, no mode if every data only appears once.)
1.2
Dispersion
1. Range:
Range = XLargest XSmallest
2. Interquartile Range:
IQR = Q3 Q1 where Q3 = 0:75(n + 1)th data and Q1 = 0:25(n + 1)th data.
If n + 1 is multiple of 4, then we take 0:75(n + 1)th data and the 0:25(n + 1)th data as
Q3 and Q1 :
If n + 1 is not multiple of 4, then we have to interpolate.
3. Variance:
PN
(Xi
)2
2
Population: = i=1
N
1
Pn
(xi x)2
n 1
4. Standard Deviation
rP
N
)2
i=1 (xi
Population: =
N
sP
n
x)2
i=1 (xi
Sample: s =
n 1
2
Sample: s =
i=1
5. Coe¢ cient of variation: CV =
1.3
s
X
100%
Relationship between two variables
1. Covariance:
Population:
XY
PN
i=1
=
Pn
(xi
X ) (yi
Y)
N
(xi x) (yi y)
Sample: sXY =
n 1
2. Correlation coe¢ cient
XY
Population: XY =
X Y
sXY
Sample: rXY =
sX sY
Note: 1
1 and 1 rXY
1
XY
i=1
2
Probability
Axiom of probability:
1. For any event E, 0 P (E) 1:
P
2. For any event E consisting outcomes O1 ; : : : ; OK , P (E) = K
i=1 P (OK ) :
3. For sample space S, P (S) = 1.
Mutually exclusive events: A and B are mutually exclusive if and only if
P (A \ B) = 0
Collectively exhaustive events: A1 , A2 ,. . . ; AK are collectively exhaustive if
P (A1 [ A2 [ : : : [ AK ) = 1
Complement rule:
P (A) = 1
P A ;P A = 1
P (A)
Odd ratio in favor of A is
Odd (A) =
P (A)
P (A)
=
1 P (A)
P A
Addition rule:
P (A \ B) = P (A) + P (B)
P (A [ B) = P (A) + P (B)
2
P (A [ B)
P (A \ B)
Conditional probability:
P (AjB) =
P (A \ B)
P (A \ B)
; P (BjA) =
P (B)
P (A)
Multiplication rule:
P (A \ B) = P (AjB) P (B) = P (BjA) P (A)
Independence: Event A and B are independent if and only if
P (A \ B) = P (A) P (B)
Note: This implies that event A and B are independent if
P (AjB) = P (A)
P (BjA) = P (B)
Law of total probability: If E1 , E2 ,. . . ; EK are mutually exclusive and collectively exhaustive, then
P (A) = P (A \ E1 ) + P (A \ E2 ) +
+ P (A \ EK )
= P (AjE1 ) P (E1 ) + P (AjE2 ) P (E2 ) +
+ P (AjEK ) P (EK )
Bayes’theorem:
3
P (BjA) =
P (AjB) P (B)
P (A)
P (Ei jA) =
P (AjEi ) P (Ei )
P (AjE1 ) P (E1 ) + P (AjE2 ) P (E2 ) +
+ P (AjEK ) P (EK )
Discrete Random Variable
For a random variable X, probability distribution function is
p (x) = P (X = x)
and the cumulative probability function:
F (x) = P (X
x) =
X
p (x)
a x
Expectation value
E (X) =
Variance:
V ar (X) =
X
X
[x
3
xp (x)
E (X)]2 p (x)
Properties of expectation and variance functions:
X
E (g (X)) =
g (x) p (x)
E (a + bX) = a + bEX
V ar (a + bX) = b2 V arX
Joint probability function for random variables X and Y
p (x; y) = P (X = x; Y = y)
Covariance
Cov (X; Y ) = E [(X E (X)) (Y E (Y ))]
XX
=
[x E (X)] [y E (Y )] p (x; y)
x
y
Correlation
Cov (X; Y )
p
V arX V arY
Properties of expectation and variance functions:
= Corr (X; Y ) = p
E (aX + bY ) = aE (X) + bE (Y )
V ar (aX + bY ) = a2 V arX + b2 V arY + 2abcov (X; Y )
4
Continuous Random Variable
For a random variable X, the cumulative probability function:
X
F (x) = P (X x) =
p (x)
a x
Hence,
P (a < X < b) = F (b)
F (a)
Properties of expectation and variance functions for random variables X and Y
E (a + bX)
V ar (a + bX)
E (aX + bY )
V ar (aX + bY )
=
=
=
=
a + bEX
b2 V arX
aE (X) + bE (Y )
a2 V arX + b2 V arY + 2abcov (X; Y )
If X is normally distributed with mean
X
and variance
N
;
2
, we denote
2
Normal distribution is symmetric about mean, bell-shaped, mean=median=mode.
Theorem: If X N ( ; 2 ), then
Z=
X
N (0; 1)
4
5
Sampling
Let X1 ; X2 ; :::; Xn be random samples from a population with mean
Sample mean
n
1X
Xi
X=
n i=1
and variance
2
.
and standard error of the mean
X
=p
n
Fact: If population is normal, then
2
X
N
;
n
Fact: If poulation is NOT normal, then by central limit theorem, when sample size is
large, then
2
X
6
N
;
n
Estimation
Consider a population parameter and its point estimator ^.
An estimator ^ is unbiased if E ^ = :
Fact: sample mean X is an unbiased estimator of population mean .
Fact: sample variance s2 is an unbiased estimator of population variance 2 .
For two unbiased estimators ^1 and ^2 , estimator ^1 is more e¢ cient if V ar ^1
V ar ^2
Interval estimators:
100(1
) % con…dence interval estimator with known
X
=2 p
Z
2
is
n
where M E = Z =2 pn , U CL = X + Z =2 pn and LCL = X Z =2 pn
100(1
) % con…dence interval estimator with unknown 2 under sample size n and is
X
where M E = tn
s
1; =2 pn ,
U CL = X + tn
tn
s
n
1; =2 p
s
1; =2 pn
5
and LCL = X
tn
s
1; =2 pn
7
Hypothesis Testing
Given population variance 2 , hypothesis testing with
sample of size n and sample mean x.
Known 2
Two-tail Test
Null Hypothesis H0
= 0
Alternative Hypothesis H1
6= 0
z =2 p
X critical values
0
n
x > 0 + z =2 p
n
Decision: Reject if
or x < 0 z =2 p
n
x
p0
test statistics
z=
= n
Z critical values
z =2
z > z =2
Decision: Reject if
or z < z =2
p-value
2P (Z > jzj)
Decision: Reject if
Without knowing population variance 2 , hypothesis
with random sample of size n, sample mean x and sample
Unknown 2
Two-tail Test
Null Hypothesis H0
= 0
Alternative Hypothesis H1
6= 0
s
X critical values
tn 1; =2 p
0
n
s
x > 0 + tn 1; =2 p
n
Decision: Reject if
s
or x < 0 tn 1; =2 p
n
x
p0
t-test statistics
t=
s= n
T critical values
tn 1; =2
t > tn 1; =2
Decision: Reject if
or t < tn 1; =2
p-value
2P (T > jtj)
Decision: Reject if
6
signi…cance level with random
Upper Tail
= 0 or
> 0
0+z p
n
x>
0
z=
+z p
x
p
0
n
Lower Tail
= 0 or
< 0
z p
0
n
x<
0
= n
+z
z>z
z p
0
z=
0
n
x
p0
= n
z
z<
z
P (Z > z)
P (Z < z)
>p
testing with
signi…cance level
standard deviation s2 :
Upper Tail
Lower Tail
= 0 or
= 0 or
0
0
> 0
< 0
s
s
tn 1; p
0 + tn 1; p
0
n
n
x>
0
+ tn
1;
x
p0
s= n
+tn 1;
t=
t > tn
1;
P (T > t)
>p
s
p
n
x<
0
t=
t<
tn
1;
x
p0
s= n
tn 1;
tn
1;
P (T < t)
s
p
n