ENGR201 FORMULA SHEET
Ch. 1
•
¯
X=x1+x2+···+xn
n=
n
P
i=1
xi
n=
m
P
i=1
xifi
m
P
i=1
fi
s2=
n
P
i=1
(xi−¯x)2
n−1=
m
P
i=1
fi(xi−¯x)2
m
P
i=1
fi−1
•Suppose that we have a data set X1,...,Xn. The pth percentile
is conceptually meant to be a value Qpsuch that p proportion
(or 100p%) of the observations fall below Qp.
Qp= the observation of rank (n+1)p(obtained by interpolation
if necessary).
•Boxplot limits
|←− 1.5IQR −→ |←− 1.5IQR −→ |←− IQR −→ |←− 1.5IQR −→ |←− 1.5IQR −→ |
Ch. 2
•The conditional probability of an event B given an event A, de-
noted as P(B|A) is
P(B|A) = P(A∩B)
P(A)
•
P(A∩B) = P(B|A)P(A) = P(A|B)P(B)
•Assume E1, E2,··· , Ekare mutually exclusive and exhaustive
events. Then the total probability of B is:
P(B) = P(B∩E1) + P(B∩E2) + ···+P(B∩Ek)
=P(B|E1)P(E1) + P(B|E2)P(E2) + ···+P(B|Ek)P(Ek)
•Two events are independent if any of the following equivalent
statements is true:
1. P(A|B) = P(A)
2. P(B|A) = P(B)
3. P(A∩B) = P(A)P(B)
•Baye’s Theorem
P(A|B) = P(B|A)P(A)
P(B)
•For a discrete random variable X with possible values
x1, x2,...,xna probability mass function is a function such that
1. f(xi)≥0
2.
n
P
i=1
f(xi)=1
3. f(xi) = P(X=xi)
•The cumulative distribution function of a discrete random vari-
able X, denoted as F(x), is
1. F(x) = P(X≤x) = P
xi≤x
f(xi)
2. 0 ≤F(x)≤1
3. If x ≤y, then F(x)≤F(y)
•The mean or expected value of the discrete random variable X,
denoted as µor E(X), is
µ=E(X) = X
x
xf(x)
The variance of X, denoted as σ2or V(X), is
σ2=V(X) = E(X−µ)2=X
x
(x−µ)2f(x) = X
x
x2f(x)−µ2
The standard deviation of X is σ=p(σ2)
•Let X be a discrete random variable with probability mass func-
tion f(x) and h(x) be any arbitrary function of X.
E[h(X)] = X
x
h(x)f(x)
•In the special case that h(X) = aX + b for any constants a and
b,
E[h(X)] = aE(X) + b
V[h(X)] = a2V(X)
•The joint probability mass function of the discrete random vari-
ables X and Y, denoted as fXY (x, y) satisfies
1. fXY (x, y)≥0
2. P
xP
y
fXY (x, y)=1
3. fXY (x, y) = P(X=x, Y =y)
•The joint probability density function for the continuous random
variables X and Y, denoted as fXY (x, y) satisfies
1. fXY (x, y)≥0 for all x,y
2. R∞
−∞ R∞
−∞ fXY (x, y)dxdy = 1
3. For any region R of two-dimensional space,
P((X, Y )∈R) = RRRfX Y (x,y)dxdy
•If the joint probability density function of random variables X
and Y is fXY (x, y), the marginal probability density functions
of X and Y are
fx(x) = Zy
fXY (x, y)dy and fy(y) = Zx
fXY (x, y)dx
where the first integral is over all points in the range of (X,Y)
for which X=x and the second integral is over all points in the
range of (X,Y) for which Y=y.
•Expected Value of a Function of Two Random Variables
E[h(X, Y )] =
PPh(x, y)fXY (x, y ) X,Y discrete
R R h(x, y)fXY (x, y )dxdy X,Y continuous
•The covariance between the random variables X and Y, denoted
as cov(X,Y) or σxy, is
σxy =E[(X−µx)(Y−µy)] = E(XY )−µxµy
•The correlation between random variables X and Y, denoted as
ρXY , is
ρXY =cov(X, Y )
pV(X)V(Y)=σXY
σXσY
•Mean of a Linear Function: If Y=c1X1+c2X2+···+cnXn,
E(Y) = c1E(X1) + c2E(X2) + ···+cnE(Xn)
•Variance of a Linear Function
If X1, X2,...,Xnare random variables, and Y=c1X1+c2X2+
···+cnXn, then in general,
V(Y) = c2
1V(X1)+c2
2V(X2)+...c2
nV(Xn)+2 X
i<j Xcicjcov(Xi, Xj)
If X1, X2,...,Xnare independent,
V(Y) = c2
1V(X1) + c2
2V(X2) + ...c2
nV(Xn)
