Edexcel A2 Statistics

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  • Created by: Imogen
  • Created on: 21-02-13 13:56

Binomial Distribution

X - B(n,p)  where n = Number of Trials, p = Probability of Success 

P(X=r) = nCr x pr x (1-p)n-r

Mean: E(X) = np 

Variance: Var(X) = np(1-p) 

Standard Deviation: √[np(1-p)] 

Criteria for using the Binomial Distribution: 

  • There must be a fixed number of trials 
  • Each trial must have the same two possible outcomes 
  • The trials must be independent on each other 
  • The probability of success must be the same for each trial 
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Poisson Distribution

X - Po(λ) 

  • The Poisson Distribution is a probability distribution. 
  • It is used when a random event occurs at a constant average rate 
  • It gives the probability for every number of events in a fixed period of time or area of space

P(X=r) = (e^-λ x λr ) / r!

Criteria for using the Poisson Distribution: 

  • Events must occur independently of one another
  • They must happen one at a time 

Finding the Mode of a Poisson Distribution:

  • If λ is a whole number then the distribution is bimodal. The modes are λ-1 and λ. 
  • If λ is not a whole number then the distribution is unimodal. The mode is the nearest whole number less than λ. 
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Poisson Distribution

Mean = E(X) = λ

Variance = Var(X) = λ

Using the Poisson Distribution to approximate the Binomial Distribution

X - B(n,p). When n is a very large value and p is close to zero, the binomial can be distributed with X - Po( λ) where λ = np. 

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Continuous Random Variables - PDF's

For a Probability Density Function:  

  • The probability that X is between a and b is the area under the curve between a and b:                                 P(a≤X≤b) =  ∫ f(x) dx with limits a and b. 
  • The probability density cannot be negative
  • The total area under the curve = 1  

∫ f(x) dx = 1

You need to be able to:

1. Recognise a PDF

  • Are any probability densities negative?
  • Is the area under the graph equal to 1?

2. Calculate probabilities

3. Define a PDF                                         

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Continuous Random Variables - CDF's

For a Cumulative Density Function, F(x) = P( X≤x)  ∫ f(x) x with limits X and ∞ With Cumulative Distribution Functions it is necessary to be able to: 

  • Calculate probabilities
  • Convert PDF's to CDF's (integration)
  • Convert CDF's to PDF's (differentiation)
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Mean and Variance of Continuous Random Variables

Expectation and Variance of a Continuous Random Variable:

Expected Value = ∫ x f(x) dx  over all possible values of x 

Variance = ∫ x² f(x) dx - E(x)   over all possible values of x

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