Extension Statistical analysis – Random variables

This lesson comprises three (3) master classes focusing on:

• Bernoulli trials
• Binomial distribution
• Normal approximation

Content:

ME-S1.1

• Use a Bernoulli random variable as a model for two-outcome situations
  • identify contexts suitable for modelling by Bernoulli random variables
• Use Bernoulli random variables and their associated probabilities to solve practical problems
  • understand and apply the formulae for the mean, \( E(X)=\overline{x}=p \), and variance, \( Var(X)=p(1−p) \), of the Bernoulli distribution with parameter \( p \), and \( X \) defined as the number of successes
• Understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in \( n \) independent Bernoulli trials, with the same probability of success \( p \) in each trial
  • calculate the expected frequencies of the various possible outcomes from a series of Bernoulli trials
• Use binomial distributions and their associated probabilities to solve practical problems
  • identify contexts suitable for modelling by binomial random variable
  • identify the binomial parameter \( p \) as the probability of success
  • understand and use the notation \( X \sim Bin(n,p) \) to indicate that the random variable \( X \) is distributed binomially with parameters \( n \) and \( p \)
  • apply the formulae for probabilities \( P(X=r)= {}^nC_r p^r(1−p)^{n−r} \) associated with the binomial distribution with parameters \( n \) and \( p \) and understand the meaning of \( ^nC_r \) as the number of ways in which an outcome with \( r \) successes can occur
  • understand and apply the formulae for the mean, \( E(X)=\overline{x}=np \), and the variance, \( Var(X)=np(1−p) \), of a binomial distribution with parameters \( n \) and \( p \)

ME-S1.2

• Use appropriate graphs to explore the behaviour of the sample proportion on collected or supplied data
  • understand the concept of the sample proportion \( \hat{p} \) as a random variable whose value varies between samples
• Explore the behaviour of the sample proportion using simulated data
  • examine the approximate normality of the distribution of \( \hat{p} \) for large samples
• Understand and use the normal approximation to the distribution of the sample proportion and its limitations