Formulas

Question 1

Binomial Coefficient

Answer 1

m!/k!(m-k)!

Question 2

Expectation of X

Answer 2

E(X) = Σxf(x)

Question 3

Variance of X

Answer 3

V(X) = E(X^2) - μ^2

Question 4

Standard Deviation of X

Answer 4

SD(X) = √σ

Question 5

Linear Transformation (Formula)
E(Y) = …
V(Y) = …
SD(Y) = …

Answer 5

Y = a + bx
E(Y) = a + bE(X)
V(Y) = b^2 V(X)
SD(Y) = I b I σ^2

Question 6

Linear Transformation (Properties)

Answer 6

1. V(a) = 0

2. E(a) = a

Question 7

Standardization of a Random Variable (Formula)

Answer 7

Z = (X-μ) / σ

Question 8

Standardization of a Random Variable (Properties)

Answer 8

E(Z) = 0
V(Z) = 1

Question 9

Chebyshev’s Inequality in Probability (Formula)

Answer 9

P(μ - kσ < X < μ + kσ) > 1- 1/(k^2)

Question 10

Bernouli Distribution (Formula)
E(X) = …
V(X) = …

Answer 10

E(X) = p
V(X) = p(1-p)

Question 11

Bernouli Distribution (Properties)

Answer 11

is used to model random variables with two outcomes

Question 12

Y ~ Bin(n,p) (Formula)

Answer 12

(nCr) (p^k) ((1-p)^n-k)

Question 13

Y ~ Bin(n,p) (Properties)
E(Y) = …
V(Y) = …

Answer 13

E(Y) = np
V(Y) = np(1-p)

Question 14

Proportion of Success (p̂) (Formula)
E(p̂) = …
V(p̂) = …

Answer 14

p̂ = #of successes/n
E(p̂) = p
V(p̂) = (p(1-p))/n

Question 15

Hypergeometric Distribution (drawn with replacement)

Y ~ Bin(n,p) where p = #successes/n
E(Y) = …
V(Y) = …

Answer 15

E(Y) = np
V(Y) = np(1-p)

Question 16

Hypergeometric Distribution (drawn without replacement)

Answer 16

Y ~ Bin(n,#successes/N) only if n/N < 0.1

Question 17

Uniform Distribution Y ~ U(α,β)
E(Y) = …
V(Y) = …

Answer 17

E(Y) = (α+ß)/2
V(Y) = (ß-α)^2/12

Question 18

Normal Distribution (Formula)
Y ~ N(μ,σ^2)
E(Y) = …
V(Y) = …

Answer 18

E(Y) = μ
V(Y) = σ^2

Question 19

Normal Distribution (3 x Properties)

Answer 19

1. pdf is symmetrical around μ
2. f(x) max at μ
3. probability = area under the graph

Question 20

Approximation of μ (Formula)

Answer 20

P(x̄ - Z(a/2) σ < μ < x̄ + Z(a/2) σ) = 1 - a

Question 21

Joint Probability Density Function
E(W) = …
COV(W) = …

Answer 21

E(W) = E(W(X,Y) = w(x,y) h(x,y)
COV(W) = E((X-E(X))(Y-E(Y))

Question 22

Expectation of a Sum
E(X) = …
V(X) = …

Answer 22

E(ΣX) = nE(X)
V(ΣX) = nV(X)

Question 23

Nominal Variables (Definition)

Answer 23

Values that cannot be ordered in a natural way

ex. (1) single, (2) married, (3) divorced, (4) widowed

Question 24

Ordinal Variables (Definition)

Answer 24

Values can be ordered

ex. (1) Poor, (2) Average, (3) Rich

Question 25

Qualitative Variables (Definition)

Answer 25

Values are categories and not numbers

Question 26

Quantitative Variables (Definition)

Answer 26

Are numbers / Variables

Question 27

Population Mean

Answer 27

(1/N) Σ X

Question 28

Population Variance

Answer 28

(1/N) Σ(X^2) - μ^2

Question 29

Sample Mean

Answer 29

(1/n) Σ X

Question 30

Sample Variance

Answer 30

s^2 = (1/n-1) Σ (x-x̄)^2

Question 31

Central Limit Theorem (Definition)

Answer 31

If n is larger than 30, then x̄ ≈ N(μ, σ^2/n)

Application only possible if np>5 and n(1-p)>5

Question 32

Approximation of p

Answer 32

P(p̂ - Z(a/2) (√(p(1-p)/n) < p < p̂ + Z(a/2) (√(p(1-p)/n)) = 1 - a

Question 33

5 Step Procedure for H0 testing

Answer 33

i) Formulate Testing Problem
ii) Test Statistic
iii) Rejection region
iv) calculate the VAL
v) draw conclusion of H0 and H1

Question 34

T Value

Answer 34

Used when σ is unkown

ta/2;n-1

Question 35

Regression Line (Formula)

Answer 35

Y = β0 + β1X + ε

Question 36

Covariance of X and Y

Answer 36

cov(X,Y) = E[(x-μx)(y-μy)]

Question 37

Expectation of X and Y

Answer 37

E(XY) = E(X)E(Y)

Question 38

Variance of a Linear Combination

W = a + bx + cx

Answer 38

V(W) = b^2V(X) + c^2V(X) + 2bcCOV(XY)