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Test #3 Formula Sheet
STAT 380
Fall 2005
Chapter 2
De Morgan’s Laws: (AB) = AB and (AB) = AB
Generalized multiplication rule: n 1
n 2
…n k
P(AB) = P(A) + P(B) – P(AB)
P(ABC) = P(A) + P(B) + P(C) – P(AB) – P(AC) – P(BC) + P(ABC)
P(A) + P(A) = 1
P(A B)
P(A | B)
P(B)
Sensitivity = P(Test is Positive | Actual is Yes)
Specificity = P(Test is Negative | Actual is No)
Chapter 3
PDF for discrete random variable properties: f(x)0, x
f(x) 1 , P(X=x) = f(x)
F(x) = P(Xx) = t x
f(t)
PDF for continuous random variable properties: f(x)0,
f(x)dx 1
, P(a<X<b) =
b
a
f(x)dx
F(x) = P(Xx) =
x
f(t)dt
d
F(x) f(x)
dx
Joint PDF for two discrete random variables properties: f(x,y) 0, x y
f(x,y) 1 , P(X=x, Y=y) =
f(x,y)
Joint PDF for two continuous random variables properties: f(x,y) 0, f(x,y) dx dy 1
P[(X, Y) A] =
A
f(x,y) dx dy
F(x,y) = P(Xx, Yy) =
x y
f(t,s) dt ds
Marginal PDF: y
g(x) f(x,y) and g(x) f(x,y) dy
Conditional PDF:
f(x,y)
f(y | x)
g(x)
for g(x) > 0
Independence: f(x,y) = g(x)h(y) and f(x|y) = g(x)
Chapter 4
= E(X) =
x
x f(x)
and = E(X) =
x f(x) dx
g(X)
= E[g(X)] =
x
g(x) f(x)
and g(X)
= E[g(X)] =
g(x) f(x) dx
g(X,Y)
= E[g(X,Y)] =
x y
g(x,y) f(x,y)
and g(X,Y)
= E[g(X,Y)] =
g(x,y) f(x,y) dx dy
Var(X) =
2
= E[(X-)
2
] =
2
x
(x ) f(x)
and Var(X) =
2
= E[(X-)
2
] =
2
(x ) f(x) dx
2
= E(X
2
2
2 2
2
g(X) g(X) g(X)
x
E g(X) g(x) f(x)
and
2 2
2
g(X) g(X) g(X)
E g(X) g(x) f(x) dx
Cov(X,Y) =
XY X Y X Y
x y
E X Y x y f(x,y)
and
XY X Y X Y
E X Y x y f(x,y) dx dy
XY
= E(XY) -
X
Y
Corr(X,Y) =
XY
XY
X Y
E[g(X) h(X)] = E[g(X)] E[h(X)]
E[g(X,Y) h(X,Y)] = E[g(X,Y)] E[h(X,Y)]
E(XY) = E(X)E(Y) if X and Y are independent
If a and b are constants, then
E(aX+b) = aE(X) + b
Var(aX + b) =
2 2 2
aX b X
a
= a
2
2
Var(aX + bY) =
2 2 2 2 2
aX bY X Y XY
a b 2ab
Var(a 1
X
1
X
2
X
n
n
2
i i
i 1
a Var(X )
+ i j i j
i j
2 a a Cov(X , X )
n
2
i i
i 1
a Var(X )
+ i j i j
i j
a a Cov(X , X )
for constants a 1
,…,a n
P( - k < X < + k) 1 -
2
k
Chapter 5
Discrete uniform PDF: 1 2 k
f(x;k) 1/ k, for x x ,x ,...,x ;
k
i
i 1
E(X) x / k
and
k
2
i
i 1
Var(X) x / k
Binomial PDF:
x n x
n
f(x;n,p) p (1 p)
x
for x=0,1,2,…,n; E(X) = np, Var(X) = np(1-p),
n n!
x x!(n x)!
Poisson PDF:
t x
e ( t)
f(x; t)
x!
for x = 0, 1, 2, …;
E(X) t and
Var(X) t
Excel functions:
BINOMDIST(x,n,p,TRUE) finds F(x) for a binomial PDF
POISSON(x, *t,TRUE)t,TRUE) finds F(x) for a Poisson PDF
Chapter 6
Uniform PDF:
for A x B
f(x;A,B) B A
0 otherwise
where
A B
E(X)
and
2
(B A)
Var(X)
Normal PDF:
2
2
(x )
2
f(x; , ) e for - x
; E(X)= and Var(X)=
2
X
Z
Chi-squared PDF:
/ 2 1 x / 2
/ 2
x e for x>
f(x) 2 ( / 2)
0 otherwise
where is a positive integer; E(X) =
and Var(X) = 2
2 2
n
i 2
2 2
i 1
(n 1)S (X X)
has a chi-squared PDF with =n-1 provided random sample from a
normal PDF
t-distribution:
( 1) / 2
2
t
h(t) 1
for - < t <
Z
T
V /
has a t-distribution provided Z is a standard normal random variable, V a chi-squared
random variable, and Z and V are independent
X
T
S / n
has a t-distribution with = n-1 provided random sample is from a normal PDF
Excel functions:
AVERAGE(range) finds the sample mean
CHIDIST(x, ) finds 1-F(x) for a chi-square PDF
CHIINV(prob., ) finds x in P(X>x) for a chi-square PDF
COUNTIF(range,”condition"
finds the # of times a condition has been satisfied
NORMDIST(x, , , TRUE) finds F(x) for a normal PDF
NORMINV(prob., , ) finds x in P(X<x) for a normal PDF
STDEV(range) finds the sample standard deviation
TDIST(t, , 1)) finds 1-H(t) for a t-distribution for a positive t
TINV(2 prob., ) finds t in P(T>t) for a t-distribution
VAR(range) finds the sample variance
Chapter 9
/ 2,n 1 / 2,n 1
s s
x t x t
n n
2
/ 2
z
n=
e
2 2 2 2
1 2 1 2
1 2 / 2, 1 2 1 2 / 2,
1 2 1 2
s s s s
x x t x x t
n n n n
where
2
2 2
1 2
1 2
2 2
2 2
1 1 2 2
1 2
s s
n n
s n s n
n 1 n 1
d d
/ 2,n 1 D / 2,n 1
s s
d t d t
n n
/ 2 / 2
2 2
/ 2 / 2
p(1 p) p(1 p)
p z p p z
n z n z
where
2
/ 2
2
/ 2
x (z ) 2
p
n (z )
2
/ 2
2
z p(1 p)
n
e
2
/ 2
2
z
n
4e
if ˆ p 0.
1 1 2 2 1 1 2 2
1 2 / 2 1 2 1 2 / 2
1 2 1 2
p (1 p ) p (1 p ) p (1 p ) p (1 p )
p p z p p p p z
n 2 n 2 n 2 n 2
where
1
1
1
x 1
p
n 2
and
2
2
2
x 1
p
n 2
n
1 n i
i 1
L(x ,...,x , ) f(x ; )
where is the parameter of interest
Chapter 10
Hypothesis test for
Test statistic:
0
x
t
s / n
Critical values for a two-tail test: ±t /2, n-
P-value for a two-tail test: 2P(T>|t|)
Hypothesis test for 1
2
2 2
1 2
1 2 1 2
1 2
s s
t x x ( )
n n
Critical values for a two-tail test: ±t /2,
where
2 2
2 2 2
2 2
1 1 2 2
1 2
1 2 1 2
s n s n
s s
n n n 1 n 1
P-value for a two-tail test: 2P(T>|t|)
Hypothesis test for D
Test statistic:
D
D
d
t
s n
Critical values for a two-tail test: ±t /2, n-
P-value for a two-tail test: 2P(T>|t|)
Hypothesis test for p
0
0 0
p p
z
p (1 p ) / n
where
p x / n
Critical values for a two-tail test: ±z
/
P-value for a two-tail test: 2P(Z>|z|)
Hypothesis test for p 1
-p 2
c c
1 2
p p 0
z
p (1 p )
n n
where 1 1 1
p x / n , 2 2 2
p x / n , and
1 2
c
1 2
x x
p
n n
Critical values for a two-tail test: ±z /
P-value for a two-tail test: 2P(Z>|z|)
