Calculus forumlae sheet for solvind differential equations and signal problems. Helps a lot for examinations.
Comments
Content
Formulae
• Addition rule: P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
P (A∩B)
P (B)
• Conditional Probability: P (A|B) =
provided P (B) > 0.
• Independence: A and B are independent if P (A ∩ B) = P (A) × P (B).
• Discrete random variable: µx =
P
x
• Continuous random variable: µx =
xp(x), σx2 =
R∞
−∞
P
x
x2 p(x) − µ2x .
xf (x)dx, σx2 =
R∞
−∞
x2 f (x) − µ2x .
• If X1 , . . . , Xn are random variables, the mean of any linear combination is given by
µc1 X1 +···+cn Xn = c1 µX1 + · · · + cn µXn .
• If X1 , . . . , Xn are independent random variables, the variance of any linear combination is given by
2
2
σc21 X1 +···+cn Xn = c21 σX
+ · · · + c2n σX
.
n
1
• Binomial distribution: p(x) =
special case with n = 1.
n
x
px (1 − p)n−x ; x = 0, 1, . . . , n. µx = np, σx2 = np(1 − p). Bernoulli is a
pr (1 − p)x−r ; x = r, r + 1, . . .. µx = pr , σx2 =
• Normal distribution: If X ∼ N (µ, σ 2 ), then Z =
X−µ
σ
r(1−p)
.
p2
∼ N (0, 1).
• Lognormal distribution: If Y ∼ LN (µ, σ 2 ), then X = loge (Y ) ∼ N (µ, σ 2 ). µy = eµ+σ
2
e2µ+σ .
• Exponential: f (x) = λe−λx , x > 0. µx =
• Uniform distribution: f (x) =
1
,
b−a
1
,
λ
σx2 =
2
/2
2
, σy2 = e2µ+2σ −
1
.
λ2
a < x < b. µx =
a+b
,
2
σx2 =
(b−a)2
.
12
• Central Limit Theorem: If X1 , . . . , Xn are independent random variables each with mean µ and standard
deviation σ, then the following hold approximately:
S = X1 + · · · + Xn ∼ N (nµ, nσ 2 )
X=
X1 + · · · + Xn
σ2
∼ N (µ,
)
n
n
provided n is large (n > 30).
• Large sample confidence interval for µ:
σ
X ± zα/2 √ .
n
Sample size needed to get a desired confidence bound B: n =
2
zα/2
σ2
B2
.
• Confidence interval for p:
r
p˜ ± zα/2
where n
˜ = n + 4, p˜ =
p˜(1 − p˜)
n
˜
X+2
.
n+4
Sample size needed for a desired confidence bound B: n =
1
2
zα/2
p∗ (1−p∗ )
B2
where p∗ is a guess of p.
• Small sample confidence interval for µ:
X ± tn−1,
s
.
n
α/2 √
• Large sample CI for µX − µY based on independent samples:
r
X − Y ± zα/2
2
σ2
σX
+ Y.
nX
nY
• CI for pX − pY :
r
p˜X − p˜Y ± zα/2
where p˜X =
X+1
,
nX +2
p˜Y =
Y +1
,
nY +2
p˜X (1 − p˜X )
p˜Y (1 − p˜Y )
+
n
˜X
n
˜Y
n
˜ X = nX + 2, n
˜ Y = nY + 2.
2
• Small-sample CI for µX − µY based on independent samples, when σX
6= σY2 :
r
X − Y ± tν,
α/2
s2X
s2
+ Y
nX
nY
where
ν=
s2
X
nX
+
/nX )2
(s2
X
nX −1
+
s2
Y
nY
2
/nY )2
(s2
Y
nY −1
,
(1)
rounded down to the nearest integer.
2
= σY2 :
• Small-sample CI for µX − µY based on independent samples, when σX
r
X − Y ± tnX +nY −2,
where
r
sp =
α/2 sp
1
1
+
nX
nY
(nX − 1)s2X + (nY − 1)s2Y
.
nX + nY − 2
• CI for paired data:
D ± tn−1,
sD
α/2 √
n
where D = X − Y .
• Large sample test for µ: Test statistic z∗ =
X−µ
√0.
σ/ n
(
P − value =
• Test for p: Test statistic z∗ = √
p−p
ˆ
0
if H1 : µ > µ0
if H1 : µ < µ0
if H1 : µ 6= µ0 .
(assume np0 > 10, n(1 − p0 ) > 10.)
p0 (1−p0 )/n
(
P − value =
• Small sample test for µ: Test statistic t∗ =
(
P − value =
P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)
P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)
if H1 : p > p0
if H1 : p < p0
if H1 : p 6= p0 .
X−µ
√0.
s/ n
P (tn−1 > t∗ )
P (tn−1 < t∗ )
2 × P (tn−1 > |t∗ |)
• Large sample test for µX − µY : Test statistic: z∗ = p
if H1 : µ > µ0
if H1 : µ < µ0
if H1 : µ 6= µ0 .
X−Y −∆0
2 /n +σ 2 /n
σX
X
Y
Y
(
P − value =
P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)
2
.
if H1 : µX − µY > ∆0
if H1 : µX − µY < ∆0
if H1 : µX − µY =
6 ∆0 .
(2)
p
ˆX −p
ˆY
• Large sample test for pX − pY : Test statistic: z∗ = √
p(1−
ˆ
p)(1/n
ˆ
X +1/nY )
pˆ =
X+Y
nX +nY
where pˆX =
X
nX
, pˆY =
Y
nY
and
.
P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)
(
P − value =
if H1 : pX > pY
if H1 : pX < pY
if H1 : pX 6= pY .
2
• Small-sample test for µX − µY , independent samples (assume σX
6= σY2 ): Test statistic: t∗ = p
P (tν > t∗ )
P (tν < t∗ )
2 × P (tν > |t∗ |)
(
P − value =
X−Y −∆0
s2
/nX +s2
/nY
X
Y
.
if H1 : µX > µY
if H1 : µX < µY
if H1 : µX 6= µY
where ν is as in (1).
2
• Small-sample test for µX − µY , independent samples (assume σX
= σY2 ): Test statistic: t∗ =
√X−Y −∆0
sp
1/nX +1/nY
,
where sp is as in (2).
(
P − value =
• Test for paired data: Test statistic: t∗ =
P (tnX +nY −2 > t∗ )
P (tnX +nY −2 < t∗ )
2 × P (tnX +nY −2 > |t∗ |)
D−µ
√0 ,
sD / n
(
P − value =
if H1 : µX > µY
if H1 : µX < µY
if H1 : µX 6= µY .
where D = X − Y .
P (tn−1 > t∗ )
P (tn−1 < t∗ )
2 × P (tn−1 > |t∗ |)
if H1 : µD > µ0
if H1 : µD < µ0
if H1 : µD 6= µ0 .
• Chi-Square test of goodness-of-fit: H0 : p1 = p10 , . . . , pk = pk0 . Test statistic:
χ2∗ =
k
X
(Oi − Ei )2
Ei
i=1
where Ei = N pi0 . P -value = P (χ2k−1 > χ2∗ ).
• Chi-Square test of homogeneity: H0 : p1j = p2j = · · · = pIj for each j (j = 1, . . . , J). Test statistic:
χ2∗ =
J
I
X
X
(Oij − Eij )2
i=1 j=1
where Eij =
Oi. O.j
O..
Eij
. P -value= P (χ2(I−1)(J−1) > χ2∗ ).
• Correlation between X and Y :
Pn
i=1
r = pPn
x2
i=1 i
xi yi − nxy
− nx2
pPn
i=1
yi2 − ny 2
• Test for ρ = ρ0 (where ρ0 6= 0): Use the test statistic W = 21 loge
1+ρ
2
1
distributed with mean µW = 21 loge 1−ρ
and variance σW
= n−3
.
.
1+r
,
1−r
which is approximately normally
√
r n−2
To test for ρ = 0, use U = √
, which has a tn−2 distribution under H0 .
2
1−r
Pn
xi yi −nxy
• Least squares regression coefficients: βˆ1 = Pi=1
, βˆ0 = y − βˆ1 x.
n
2
2
i=1
xi −nx
• 100(1 − α)% CI for β0 and β1 are: βˆ0 ± tn−2,α/2 sβˆ0 and βˆ1 ± tn−2,α/2 sβˆ1 where
s
sβˆ0 = s
1
x2
+ Pn
,
n
(xi − x)2
i=1
sβˆ1 = pPn
i=1
3
s
(xi − x)2
r
and
s=
(1 − r2 )
Pn
(yi − y)2
.
n−2
i=1
• 100(1 − α)% CI for the mean predicted value at x is βˆ0 + βˆ1 x ± tn−2,α/2 syˆ where
s
syˆ = s
(x − x)2
1
+ Pn
n
(xi − x)2
i=1
• 100(1 − α)% prediction interval at x is given by βˆ0 + βˆ1 x ± tn−2,α/2 spred where
s
spred = s
1+
(x − x)2
1
+ Pn
n
(xi − x)2
i=1
Pn
• Regression SS (SSR) = i=1 (ˆ
yi − y)2 , Error SS (SSE) =
Analysis of variance identity: SST = SSR + SSE.
• s2 =
SSE
.
n−p−1
Coefficient of determination R2 =
• To test H0 : β1 = · · · = βp = 0, use F =
Pn
i=1
(yi − yˆi )2 and Total SS (SST) =
SSR
.
SST
SSR/p
.
SSE/(n−p−1)
Under H0 , F ∼ Fp,n−p−1 .
• F -test for one-way ANOVA:
SST r =
I
X
2
2
Ji X i. − N X ..
i=1
SSE =
Ji
I
X
X
I
X
2
Xij
−
i=1 j=1
2
Ji X i.
i=1
SST r
SSE
M ST r
M ST r =
, M SE =
, F =
I −1
N −I
M SE
Under H0 : µ1 = · · · = µI , F ∼ FI−1, N −I .
• 100(1 − α)% CI for µi is X i. ± tN −I,
q
α/2
M SE
Ji
• Fisher’s Least Significant Difference Method:
– 100(1 − α)% CI for (µi − µj ) is X i. − X j. ± tN −I,α/2
q
M SE( J1i +
1
Jj
)
– To test H0 : µi = µj , reject H0 at level α if
r
|X i. − X j. | > tN −I,α/2
M SE(
1
1
+ )
Ji
Jj
• Bonferroni Method with C simultaneous comparisons:
– 100(1 − α)% CI for (µi − µj ) is X i. − X j. ± tN −I,α/(2C)
q
M SE( J1i +
1
Jj
)
– To test H0 : µi = µj , reject H0 at level α if
r
|X i. − X j. | > tN −I,α/(2C)
M SE(
1
1
+ )
Ji
Jj
• Tukey-Kramer Method for all possible comparisons:
– 100(1 − α)% CI for (µi − µj ) is X i. − X j. ± qI,N −I,α