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
