List of Formul
ST370 - Probability and Statistics for Engineers
(bring this list for all exams)
I. Exploring statistical data and regression analysis.
1. X ; X ; ; Xn :: Sample data consisting of n observations.
2. X = n Pni Xi:: Sample mean.
3. Qd = X k + [(n + 1)d , k](X k , X k ):: Sample d-th fractile/quantile,
where k =largest integer (n + 1)d and X X : : : X n represents
data sorted in ascending order. Notice that the median, m = Q : .
Pni=1 Xi,X 2 P Xi2,nX 2
= n, :: Sample variance.
4. s =
p n,
5. s = s : sample standard deviation (sd).
6. R = X n , X :: Sample range.
7. IQR = Q : , Q : :: Inter-quartile range.
8. = Xs :: Coecient of variation.
9. SK = Xs,m :: Skewness coecient. SK = 0 ) symmetric, SK > 0 )
+vely skewed and SK < 0 ) ,vely skewed.
Pni=1 Xi,X Yi,Y P XiYi ,nXY
10. r =
= , sX sY :: sample correlation coecient.
n, sX sY
Pni=1 XiYi ,nnXY
11. b = rsY =sX =
:: The slope when Y is regressed on X (i.e.
n, s2X
Y a + bX ).
12. a = Y , bX : The vertical intercept when Y is regressed on X.
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13. Yd
(x) = a + bx : Predicted value of Y when X=x is observed.
r Pn
14. sY:X = i=1n,Yi ,Ybi : Standard error of the estimate when Y is regressed
on X .
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15. R = 1 , nn,, sY:X
s2Y :: Multiple R-squared (Coecient of determination).
Note R = r only for Linear regression.
16. ei = Yi , Y d
(Xi):: Residuals when Y is regressed on X .
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c Sujit K. Ghosh, NC State University
ST 370 List of formul.
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II. Probability and random variables (r.v.).
1. Pr[not A] = 1 , Pr(A):: Probability of a complementary event.
2. Pr[A and B ] = 0:: Implies A and B are mutually exclusive or disjoint
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
events.
Pr[A or B ] = Pr[A] + Pr[B ] , Pr[A and B ]:: General Addition law of probability.
Pr[A j B ] = Pr[A and B ]= Pr[B ]:: Conditional probability of A given B .
Pr[A j B ] = Pr[A] or Pr[A and B ] = Pr[A] Pr[B ]:: Implies A and B are
statistically independent events.
Pr[A and B ] = Pr[A j B ] Pr[B ] = Pr[B j A] Pr[A] : General Multiplication
law of probability.
p(y) = Pr[Y = y]; P1y p(y) = 1:: Probability mass function of a discrete
random variable Y. Note 0 p(y) 1, where y = 0; 1; : : :.
E (X ) = P xp(x):: Expected value of a discrete random variable X .
V ar(X ) =qP[x , E (X )] p(x): : Variance of a discrete random variable X .
SD(X ) = V ar(X ); standard deviation of X .
R 1 f (x)dx = 1:: Probability density function of continuous
f (x) 0; ,1
random variable X.
R 1 xf (x) dx:: Expected value of a continuous random variable
E (X ) = ,1
X.
1 [x , E (X )] f (x) dx:: Variance of a continuous random variV ar(X ) = R,1
q
able X . SD(X ) = V ar(X ); standard deviation of X .
E (a + bX ) = a + bE (X ):: Property of expected value (holds for both discrete
and continuous r.v.).
V ar(a + bX ) = b V ar(X ):: Property of variance (holds for both discrete and
continuous r.v.). Note that V ar(X ) = E (X ) , [E (X )] :
F (x) = Pr[X x]:: Cumulative probability distribution function of X . Notice that Pr[a < X b] = F (b) , F (a):
=0
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c Sujit K. Ghosh, NC State University
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ST 370 List of formul.
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16. b(r; n; ) = Crnr (1 , )n,r = Pr[R = r]:: Binomial probability mass function, where R is the number of success out of n trials and Pr[success] = .
17. E (R) = n; V ar(R) = n(1,):: Expected value and variance for a Binomial
Distribution.
18. B (r; n; ) = Pr[R r] = Prx b(x; n; ):: Binomial probability distribution
function. (see Table A)
19. f (x) = p expf, x,2 2 g:: Probability Density function of a Normal distribution.
20. E (X ) = ; V ar(X ) = :: Expected value and variance for a normal distribution.
21. Z = X, :: Normal deviate. Notice that E (Z ) = 0 and V ar(Z ) = 1.
R z p e, y22 dy:: Standard normal probability distri22. (z) = Pr[Z z] = ,1
bution function. (see Table D)
,t x
23. p(x; ; t) = Pr[X = x] = e x t ; x = 0; 1; 2 : : : :: Probability mass function
for Poisson distribution.
24. P (x; ; t) = Pr[X x] = Pxk p(k; ; t):: Poisson probability distribution
function. (see Table C)
25. E (X ) = t = V ar(X ):: Expected value and variance for a Poisson Distribution.
26. f (t) = e,t; t 0:: Probability density function for the Exponential distribution.
27. F (t) = Pr[T t] = 1 , e,t:: Exponential probability distribution function.
28. E (T ) = ; V ar(T ) = 2 :: Expected value and variance for an Exponential
Distribution.
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c Sujit K. Ghosh, NC State University
ST 370 List of formul.
3
III. Sampling distribution and statistical inference.
1. E (X ) = ; SD(X ) = X = pn : Property q
of sample mean for large population. Note for small population, X = pn NN,,n :
2. Central Limit Theorem (CLT): The sampling distribution
of X approaches
2
a normal curve with expected value and variance n as n becomes large,
regardless of the form of population frequency distribution.
3. X z= pn :: Condence interval estimate of , when known. (see Table
D or E for z= )
4. X t= psn :: Condence interval estimate of , when unknown. (see
Table G for t= with df = n , 1)
z2 2
5. n = =d22 : : Required sample size for estimating the mean, where d =
desired precision, z= =critical normal deviate and = assumed population
sd. If is unknown, replace it by its estimate say R=6 (recall, R = X n ,X ).
q
6. P z= P n,P :: Condence interval estimate of the proportion when
the population is large. This estimate is based on the assumption that n 5
and n(1 , ) 5.
2
7. n = z=2 d2 , : : Required sample size for estimating the proportion, where
d = desired precision, z= =critical normal deviate and = assumed population proportion. If is unknown, replace it by a conservative estimate,
= 0:5.
r s2 s2
8. (XA , XB ) z= nAA + nBB :: Condence interval estimate for A , B using
large independent samples (min(nA; nB ) > 30).
9. (XA ,XB )t= sD :: Condence interval estimate for A ,B using small independent
samples (min(nA ; nB ) r30).
2
2q
When A = B ; sD = nA , nAsA nBn,B , sB nA + nB . (see Table G for t=
with df = nA + nB , 2)
r s2 s2
When A 6= B ; sD = nAA + nBB : (see Table G for t= with
2
2
2
df = s2A=nA 2= snAA=n,A sBs2B=n=nBB 2= nB , )
10. d t= psdn :: Condence interval estimate of A , B using matched pairs.
(see Table G for t= with df = n , 1)
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c Sujit K. Ghosh, NC State University
ST 370 List of formul.
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