List of formulae and tables of the normal distribution (MF9)
6.
List of formulae and tables of the normal distribution
(MF9)
PURE MATHEMATICS
Algebra
For the quadratic equation ax2 + bx + c = 0:
x=
For an arithmetic series:
u n = a + ^ n − 1 h d,
S n = 12 n ^a + l h = 12 n "2a + ^n − 1h d ,
For a geometric series:
u n = arn − 1,
− b ! : ^b2 − 4ach
2a
Sn =
a ^1 − rnh
^r ! 1h,
1−r
a ^ r < 1h
S3 = 1 −
r
Binomial expansion:
Kn O
Kn O
Kn O
^a + bhn = an + KK OO an − 1 b + KK OO an − 2 b2 + KK OO an − 3 b3 + g + bn, where n is a positive integer
J N
J N
J N
1
2
3
L P
L P
L P
JKn NO
n!
and KK OO =
^n − r h !
r
!
r
L P
n ^n − 1h 2 n ^n − 1h^n − 2h 3
^1 + xhn = 1 + nx +
+
x g, where n is rational and x < 1
2! x
3!
Trigonometry
Arc length of circle = ri ^i in radiansh
Area of sector of circle = 12 r2 i ^i in radiansh
sin i
cos i
1 + tan2 i / sec2 i,
tan i /
cos2 i + sin2 i / 1.
cot2 i + 1 / cosec2 i
sin ^ A ! Bh / sin A cos B ! cos A sin B
cos ^ A ! Bh / cos A cos B " sin A sin B
A ! tan B
tan ^ A ! Bh / 1tan
" tan A tan B
sin 2A / 2 sin A cos A
2
cos 2A / cos A − sin2 A / 2 cos2 A − 1 / 1 − 2 sin2 A
tan 2A = 2 tan A
1 − tan2 A
Principal values:
− 12 r G sin−1 x G 12 r
0 G cos−1 x G r
− 12 r < tan−1 x < 12 r
28
Cambridge International AS and A Level Mathematics 9709. Syllabus for examination in 2017 and 2018.
List of formulae and tables of the normal distribution (MF9)
Differentiation
f^ xh
xn
1n x
ex
sin x
cos x
tan x
uv
f l^ x h
nxn–1
1
x
ex
cos x
–sin x
sec2 x
u ddvx + v ddux
v ddux − u ddvx
u
v
v2
dy dy
If x = f (t) and y = g (t) then dx = dt ' ddxt
Integration
f^ xh
xn
1
x
ex
sin x
cos x
sec2 x
y f (x) dx
x n + 1 + c ^n ! − 1 h
n+1
1n x + c
ex + c
– cos x + c
sin x + c
tan x + c
y u ddvx dx = uv − y v ddux dx
y ffl^^xxhh dx = 1n f^ xh + c
Vectors
If a = a1i + a2j + a3k and b = b1i + b2j + b3k then
a.b = a1b1 + a2b2 + a3b3 = a b cos i
Numerical integration
Trapezium rule:
b
y f ^ xhdx .
a
1
2
−
h " y0 + 2 ^ y1 + y 2 + g + y n − 1h + y n ,, where h = b n a
Cambridge International AS and A Level Mathematics 9709. Syllabus for examination in 2017 and 2018.
29
List of formulae and tables of the normal distribution (MF9)
MECHANICS
Uniformly accelerated motion
v = u + at,
s = 12 ^u + vh t,
s = ut + 12 at2,
v2 = u2 + 2as
Motion of a projectile
Equation of trajectory is:
y = x tan i −
Elastic strings and springs
gx2
2V cos2 i
T = mlx ,
2
2
E = m2xl
Motion in a circle
For uniform circular motion, the acceleration is directed towards the centre and has magnitude
v2
~2 r or
r
Centres of mass of uniform bodies
Triangular lamina:
2
3
along median from vertex
Solid hemisphere of radius r: 83 r from centre
Hemispherical shell of radius r: 12 r from centre
Circular arc of radius r and angle 2a:
r sin a
a from centre
Circular sector of radius r and angle 2a:
2r sin a
3a from centre
Solid cone or pyramid of height h: 34 h from vertex
30
Cambridge International AS and A Level Mathematics 9709. Syllabus for examination in 2017 and 2018.
List of formulae and tables of the normal distribution (MF9)
PROBABILITY AND STATISTICS
Summary statistics
For ungrouped data:
/x
x= n ,
/ ^ x − x h2
standard deviation =
n
=
/ x2
n −x2
For grouped data:
/ xf
x= / ,
f
/ ^ x − x h2 f
=
/f
standard deviation =
Discrete random variables
/ x2 f
2
/ f −x
E ^ X h = / xp
Var ^ X h = / x2 p − "E ^ X h, 2
For the binomial distribution B(n, p):
JKn NO
p r = KK OO pr ^1 − phn − r,
r
L P
For the Poisson distribution Po(a):
r
p r = e−a ar! ,
n = np,
v2 = np ^1 − ph
n = a,
v2 = a
Continuous random variables
E ^ X h = xf ^ x h dx
y
Var ^ X h = x2 f ^ x h dx − "E ^ X h, 2
y
Sampling and testing
Unbiased estimators:
/x
x= n ,
JK
^/ xh NO
s2 = n 1− 1 K/ x2 − n O
P
L
2
Central Limit Theorem:
v2
X + N ` n, n j
Approximate distribution of sample proportion:
JK p ^1 − phNO
N K p,
n OP
L
Cambridge International AS and A Level Mathematics 9709. Syllabus for examination in 2017 and 2018.
31
List of formulae and tables of the normal distribution (MF9)
32
Cambridge International AS and A Level Mathematics 9709. Syllabus for examination in 2017 and 2018.