Statistics formula sheet
Summarising data
Sample mean: x= Sample variance: s2 x 1 = n−1 n This has mean nθ and variance nθ(1 − θ). The Poisson distribution: p(x) = λx exp(−λ) for x = 0, 1, 2, . . . . x!

1 n

n

This has mean λ and variance λ. xi .

i=1

Continuous distributions n Distribution function: x2 i − nx
2

i=1

1 (xi − x) = n−1
2

. F (y) = P (X ≤ y) =

y

i=1

f (x) dx.
−∞

Sample covariance: g= 1 n−1 n Density function: 1 n−1 n (xi −x)(yi −y) = i=1 xi yi − nx y i=1 .

f (x) = Evaluating probabilities:

d F (x). dx

Sample correlation: r= g . sx sy

b

P (a < X ≤ b) = a f (x) dx = F (b) − F (a).

Probability
Addition law: P (A ∪ B) = P (A) + P (B) − P (A ∩ B). Multiplication law: P (A ∩ B) = P (A)P (B|A) = P (B)P (A|B). Partition law: For a partition B1 , B2 , . . . , Bk k k

Expected value:
∞

E(X) = µ =
−∞

xf (x) dx.

Variance:
∞ ∞

Var(X) =
−∞

(x − µ)2 f (x) dx =
−∞

x2 f (x) dx − µ2 .

Hazard function: h(t) = P (A|Bi )P (Bi ). i=1 f (t) . 1 − F (t)

P (A) = i=1 P (A ∩ Bi ) =

Normal density with mean µ and variance σ 2 : 1 f (x) = √ exp 2πσ 2 . Weibull density: f (t) = λκtκ−1 exp(−λtκ ) for t ≥ 0. Exponential density: − 1 2 x−µ σ
2

Bayes’ formula: P (A|Bi )P (Bi ) P (Bi |A) = = P (A) P (A|Bi )P (Bi ) k i=1

for x ∈ [−∞, ∞].

P (A|Bi )P (Bi )

Discrete distributions
Mean value: E(X) = µ = xi ∈S

f (t) = λ exp(−λt) for t ≥ 0. xi p(xi ). This has mean λ−1 and variance λ−2 .

Variance: Var(X) = xi ∈S

Test for population mean
(xi − µ)2 p(xi ) = xi ∈S

x2 p(xi ) − µ2 . i Data: Single sample of measurements x1 , . . . , xn . Hypothesis: H : µ = µ0 . Method: √ • Calculate x, s2 , and t = |x − µ0 | n/s. • Obtain critical value from t-tables, df = n − 1.

The binomial distribution: p(x) = n x θ (1 − θ)n−x for x = 0, 1, . . . , n. x