S2 Notes (Edexcel)

S2 EDEXCEL REVISION NOTES

Binomial Distribution
Binomial probability distribution is defined as: * P(X=r) = nCr x pn x (1-p)n-r * Distribution is written as: X~B(n,p)
Conditions include: * Fixed number of trials * All trials are independent of one another * Probability of success remains constant * Each trial much have the same two possible outcomes
E(X) = np
Var(X) = npq [where q = 1 - p]
SD(X) = Var (X) = npq
To calculate the probabilities: * P(X ≤ x) = read off the tables * P(X ≥ x) = 1- P(X ≤ x-1) * P(X < x) = P(X ≤ x-1) * P(X > x) = 1- P(X ≤ x) * P(x ≤ X ≤ y) = P(X ≤ y) – P(X ≤ x-1) * P(x < X < y) = P(X ≤ y) – P(X ≤ x) * P(x ≤ X < y) = P(X ≤ y-1) – P(X ≤ x-1) * P(x < X ≤ y) = P(X ≤ y) – P(X ≤ x)
Because...
X changes to Y
NEW value for n is 18 – 8 =10
NEW value for p is 1.0 – 0.9 = 0.1
And the sign swaps round
We can now read this off the tables

Note: If p > 0.5, need to make X~B(n,p) convert to Y~B(n,p)
Example:
If X~B(18,0.9), 0.9 is > 0.5 therefore not on the tables
We want to find P(X > 8)
This will become P(Y < 10) AND Y~B(10,0.1)

Poisson Distribution
Binomial probability distribution is defined as: * P(X=r) = e-λ x λrr! * Distribution is written as: X~Po(λ)
Conditions include: * Events occur at random * All events are independent of one another * Average rate of occurrence remains constant * Zero probability of simultaneous occurrences
E(X) = λ
Var(X) = λ
SD(X) = Var (X) = λ
To calculate the probabilities: * P(X ≤ x) = read off the tables * P(X ≥ x) = 1- P(X ≤ x-1) * P(X < x) = P(X ≤ x-1) * P(X > x) = 1- P(X ≤ x) * P(x ≤ X ≤ y) = P(X ≤ y) – P(X ≤ x-1) * P(x < X < y) = P(X ≤ y) – P(X ≤ x) * P(x ≤ X < y) = P(X ≤ y-1) – P(X ≤ x-1) * P(x < X ≤ y) = P(X ≤ y) – P(X ≤ x)
To approximate the poisson to the binomial, the following