S2 Flashcards
Binomial Distribution
Conditions
Fixed number of trials
Independent trials
Constant probability of success
Two possible outcomes
Binomial Distribution
Notation
X~B (n,p)
n = number of trials p = probability of success
Binomial Distribution
Formula
P(X = r) = nCr x p^r x q^n-R
Binomial Distribution
Expectation
E(X) = np
Binomial Distribution
Variance
Var(X) = npq
Binomial Distribution
Poisson Approximation
If n is large and p is small
Then
X~B (n,p) ≈ Y~Po(np)
Binomial Distribution
Normal Approximation
If n is large, np>5 and nq>5
Then
X~B (n,p) ≈ Y~N (np,npq)
Using a half continuity correction
Poisson Distribution
Notation
X~Po (λ)
Poisson Distribution
Conditions
Events occur independently, singly and at a constant rate
Poisson Distribution
Formula
P(X =r) = (e^-λ x λ^r) / r!
Poisson Distribution
Expectation
E(X) = λ
Poisson Distribution
Variance
Var(X) = λ
Poisson Distribution
Normal Approximation
If λ>10
Then
X~Po (λ) ≈ Y~N (λ,λ)
Probability Density Function
Properties
f(x) ≥ 1 for all values of x
The area under the line = 1
Notation
f(x) =
function of x, a≤x≤b
0, otherwise
Cumulative Distribution Function
Properties
0 ≤ F(x) ≤ 1
Area at start = 0
Area at end = 0
CDF
Notation
F(x) =
0, a<b
CDF to PDF
Differentiate
PDF to CDF
Integrate
Remember c
Continuous Random Variable
Mode
Maximum value of f(x)
Continuous Random Variable
Median
F(m) = 0.5
Continuous Random Variable
Mean
Multiply PDF by x then integrate between the bonds
Continuous Random Variable
Variance
(Multiply PDF by x² then integrate between the bounds) - mean²
Continuous Uniform Distribution
1 / b-a
Continuous Uniform Distribution
PDF shape
Straight horizontal line
Continuous Uniform Distribution
Median
(a+b) / 2
Continuous Uniform Distribution
Mean
(a+b) / 2
Continuous Uniform Distribution
Variance
(b - a)² / 12
Sampling
Benefits
Representative
Useful when the test would harm or damage the item
Cheap
Quick
Sampling
Restrictions
Natural variation
Bias
Population
Definition
Collection of individuals or items
Finite Population
Definition
A population for which the number of members can be known
Infinite Population
Definition
A population for which the number of members cannot be known
Census
Definition
Taking information from every member of the population
Sample
Definition
A selection of members form the population
Can be any size
Sampling Unit
Definition
A member of the sample
Sample Frame
Definition
List of all the possible sampling units in the population
Statistic
Definition
A function of the observations and no other values
Critical Region
Definition
The range of values for which the null hypothesis is rejected
Significance Level
Definition
The probability used for very unlikely
