Advanced Statistical analysis – Random variables

This lesson comprises two (2) master classes focusing on:

• Continuous random variables
• Probability density function
• Cumulative distribution function
• Normally distributed random variables
• z-scores
• Empirical rule
• Using z-scores to compare data sets
• Using z-scores to calculate probabilities

Content:

MA-S3.1

• Use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable
• Understand and use the concepts of a probability density function of a continuous random variable
  • know the two properties of a probability density function: \( f(x) \ge 0 \) for all real \( x \) and \( \int_{-\infty}^{\infty}f(x)dx=1 \)
  • define the probability as the area under the graph of the probability density function using the notation \( P(X \le r)=\int_a^r f(x)dx \), where \( f(x) \) is the probability density function defined on \( [a,b] \)
  • examine simple types of continuous random variables and use them in appropriate contexts
  • explore properties of a continuous random variable that is uniformly distributed
  • find the mode from a given probability density function
• Obtain and analyse a cumulative distribution function with respect to a given probability density function
  • understand the meaning of a cumulative distribution function with respect to a given probability density function
  • use a cumulative distribution function to calculate the median and other percentiles

MA-S3.3

• Identify the numerical and graphical properties of data that is normally distributed
• Calculate probabilities and quantiles associated with a given normal distribution using technology and otherwise, and use these to solve practical problems
  • identify contexts that are suitable for modelling by normal random variable, eg the height of a group of students
  • recognise features of the graph of the probability density function of the normal distribution with mean \( \mu \) and standard deviation \( \sigma \), and the use of the standard normal distribution
  • visually represent probabilities by shading areas under the normal curve, eg identifying the value above which the top 10% of data lies
• Understand and calculate the z-score (standardised score) corresponding to a particular value in a dataset
  • use the formula \( z=\frac{x− \mu}{\sigma} \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation
  • describe the z-score as the number of standard deviations a value lies above or below the mean
• Use z-scores to compare scores from different datasets, for example comparing students’ subject examination scores
• Use collected data to illustrate the empirical rules for normally distributed random variable
  • apply the empirical rule to a variety of problems
  • sketch the graphs of \( f(x)=e^{−x^2} \) and the probability density function for the normal distribution \( f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{−\frac{(x−μ)^2}{2 \sigma ^2}} \) using technology
  • verify, using the Trapezoidal rule, the results concerning the areas under the normal curve
• Use z-scores to identify probabilities of events less or more extreme than a given event
  • use statistical tables to determine probabilities
  • use technology to determine probabilities 
• use z-scores to make judgements related to outcomes of a given event or sets of data