# S2

Short synopsis of S2 Edexcel

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## Binomial Distribution

If X-B(n,p):

• P(X=x) = (nCx)*(p^x)*(1-p)^(n-x)
• E(X) = np
• Var(X) = np(1-p)

Condions Needed:

• Fixed number of trials
• Each trial only has two out comes(e.g. success or failure)
• Trials should be independent
• The probabability of success stays constant

Can be Approximated to the Poisson distribution when n is large and p is small

ƛ = np

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## Poisson Distribution

If X≈Po(ƛ)

• P(X=x) = (e^-ƛ)*(ƛ^x)/x!
• E(X) = Var(X) = ƛ
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## Continous Random Variables

• E(X) = integrate xf(x)
• Var(X) = E(X^2) = integrate x*x*f(x) - E(X)^2
• The mode is the highest point of the function
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## Continuous Uniform Distribution

A variable is distributed over the interval (a,b)

The p.d.f. is = { 1/(b-a) , a<x<b

0, Otherwise

E(X) = (a+b)/2

Var(X) = (b-a)^2/12

F(X) = { 0, x<a

(x-a)/(b-a) , a<x<b

1, x>b

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## Normal Approximations

When n is large and p is close to 0.5, the binomial distribution can be approximated by the normal distribution, where y-N(np,np(1-p))

The poisson distribution can be approximated by the normal distribution when ƛ is large where y-N(ƛ,ƛ)

Continuity Correction is needed for both these approximations

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## Populations and Samples

Population: A collection of individual items

Sample: A selection of individual items from a population

Finite Population: A population where each item cane given a number

Infinite Population: A population where numbering each member is impossible

Sampling Unit: An individual member of a population

Sampling Frame: A list of sampling units used in practice to represent a population

Statistic: A quantity calculated solely from the observations in a sample

Sampling Distribution: A statistic that is defined by giving all possible values of the statistic and the probability of each occurring

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## Hypothesis Testing

H0 = Null hypothesis: this represents the the original hypothesis

H1 = Alternate hypothesis: this represents the proposed hypothesis

Once the calculations are done if the value lies within the significance region then there is enough evidence to prove H1 correct, otherwise you say there isn't enough evidence.

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## Continuity Correction

P(X=b) --> P(b-1/2 < X < b+1/2)

P(X<=b) --> P(X<b+1/2)

P(X<b) --> P(X<b-1/2)

P(X>=b) --> P(X>b-1/2)

P(X>b) --> P(X>b+1/2)

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