ADM2303 formula sheet: 2023
Random variables (RV)
Discrete and continuous distributions
Expected value of discrete RV X :
The Binomial probability distribution:
Probability theory
E(X) = µ =
Rule of sum of probabilities:
n
X
xi P(X = xi )
i=1
P(S) = 1
P(A) = 1 − P(Ac )
=
n
X
(xi − µ)2 P(X = xi )
i=1
=
n
X
2
x2i P(X = xi ) − µ2
i=1
Addition rule for two mutually exclusive events (where ∪
connotes “or” aka union):
Standard deviation of discrete RV X:
p
SD(X) = σ = V ar(X)
P(A ∪ B) = P(A) + P(B)
Coefficient of variation of discrete RV X:
Addition rule for two not mutually exclusive events:
CV (X) =
SD(X)
E(X)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Multiplication rule for two independent events (where ∩
connotes “and” aka intersection):
Correlation of two discrete RV X and Y : ρx,y
Pn
i=1 (xi − µx )(yi − µy ))P(X = xi ∩ Y = yj )
=
sx sy
Combining random variables
P(A1 ∩ A2 ∩ ...An ) = P(A1 ) × P(A2 ) × ... × P(An )
Multiplication rule for dependent events:
P(A ∩ B) = P(B|A)P(A) = P(A|B)P(B)
Partition rule: for a partition B1 , B2 , ..., Bk :
P(A) =
k
X
P(A ∩ Bi ) =
i=1
k
X
The Poisson probability distribution
(if approx’n of binomial, λ = np):
e−λ λx
for x = 0, 1, 2, ...
x!
E(X) = λ, V ar(X) = λ, e = 2.718
P(X = x) =
The Geometric probability distribution:
P(X = x) = (1 − p)x−1 p for x = 1, 2, ...
1−p
1
E(X) = , V ar(X) =
p
p2
The Normal distribution:
P(A ∩ B) = P(A) × P(B)
Multiplication rule for n independent events:
n!
px (1 − p)n−x for x = 0, 1, ..., n
(n − x)!x!
E(X) = np, V ar(X) = np(1 − p)
Variance of discrete RV X: Var(X) = σ
Complement rule
(Let Ac be complement of A, i.e., Not A):
P(X = x) =
Adding a constant c to random variable X:
E(X ± c) = E(X) ± c
1
1 x−µ 2
X ∼ N (µ, σ) ⇒ f (x) = √ exp − (
)
2
σ
σ 2π
X −µ
Z=
∼ N (0, 1) ⇒ P(Z<z) = using normal table
σ
2
V ar(X ± c) = V ar(X) = σX
Multiplying random variable X by a constant a :
E(aX) = aE(X)
The Exponential distribution :
X ∼ Expo(λ) ⇒ f (x) = λe−λx
P(X ≤ a) = 1 − e−aλ
1
1
E(X) = , V ar(X) = ( )2
λ
λ
2
V ar(aX) = a2 σX
P(A|Bi )P(Bi )
i=1
Expected value of linear combination of RVs:1
Bayes’ formula:
E(aX + bY + c) = aE(X) + bE(Y ) + c
P(A|Bi )P(Bi )
P(A|Bi )P(Bi )
= Pk
P(Bi |A) =
P(A)
i=1 P(A|Bi )P(Bi )
Events A and B are independent if:
P(A|B) = P(A) and P(B|A) = P(B)
or:P(A ∩ B) = P(A) × P(B)
1 For cases like 2X − 3Y
The Uniform distribution:
Variance of linear combination of RVs(1) :
2
2
V ar(aX + bY + c) = a σX
+ b2 σY2 + 2 a b Cov(X, Y )
where Cov(X, Y ) = ρx,y σX σY . If X and Y independent
then ρx,y = 0 and covariance component drops out.
1
for a ≤ x ≤ b
b−a
x2 − x1
P(x1 <X<x2 ) =
b−a
(b − a)2
a+b
E(X) =
, V ar(X) =
2
12
X ∼ Uniform(a, b) ⇒ f (x) =
the coefficient on Y is (−3), thus treat accordingly; want to subtract a constant rather than add — put minus sign on c.
Sampling distributions for proportion
Descriptive statistics
Sample mean:
x̄ =
n
1X
n i=1
X
n
If n is large i.e. np ≥ 10 and n(1 − p) ≥ 10
!
r
p(1 − p)
⇒ p̂ ∼ N p,
n
X ∼ Binomial(n, p), and p̂ =
xi
Sample variance:
n
1 X
1
(xi − x̄)2 =
n − 1 i=1
n−1
s2x =
n
X
!
x2i − nx̄2
Sampling distributions for mean
i=1
σ
If X ∼ N (µ, σ) ⇒ X̄ ∼ N (µ, √ )
n
Sample coefficient of variation:
CV =
s
If X ∼ N (µ, unknown) ⇒ X̄ ∼ tdf =n−1 (µ, √ )
n
σ
If n is Large
If X ∼ unknown(µ, σ) ========⇒ X̄ ∼ N (µ, √ )
CLT
n
s
x̄
Sample covariance:
n
1 X
1
Cov(x,y) =
(xi − x̄)(yi − ȳ) =
n − 1 i=1
n−1
n
X
!
xi yi − nx̄ȳ
i=1
Sample correlation:
s
If n is Large
If X ∼ unknown(µ, unknown) ========⇒ X̄ ∼ tdf =n−1 (µ, √ )
CLT
n
Finite population correction factor
n
> 10% use :
In case of a finite population where N
r=
Cov(x, y)
=
sx sy
Pn
i=1 (xi − x̄)(yi − ȳ)
(n − 1)sx sy
Percentile:
- sort your data first
kth percentile index : i = (
k
)(n + 1)
100
- if i is integer, kth percentile is the ith value
- if i is not integer, kth percentile is mean of the observations on either side of i
Boxplot elements:
IQR = Q3 − Q1
U pperLimit = Q3 + 1.5(IQR)
LowerLimit = Q1 − 1.5(IQR)
Normal approximation to Binomial
If X ∼ Binomial(n, p)
If n is large i.e. np ≥ 10 and n(1 − p) ≥ 10
p
⇒ X ∼ N µx = np, σx = np(1 − p)
r
standard deviation ×
N −n
N −1