Random Variables

statisticalanalysis year12 advanced

Random variables | NSW Curriculum Website

• MAV-12-07 solves problems involving discrete probability distributions, continuous random variables and the normal distribution

📖 Prior Knowledge

Content	Prior knowledge	Used for
Data Classification and Visualisation	- frequency tables and histograms	- probability distributions
Data Analysis	- mean - summary statistics, shape of a dataset	- expected value - identifying normal distributions
Probability	- relative frequency	- estimating probabilities of a discrete random variable
Data Analysis A	- standard deviation	- variance and standard deviation - empirical rule

Discrete random variables

• Use the relative frequencies of discrete random variable datasets to estimate the probabilities that the random variable $X$ takes each of its values, and explain why these estimated probabilities add to 1
• Denote the probability that the discrete random variable $X$ takes the value $x$ by $P(X=x)$, or by $P(x)$ if the discrete random variable $X$ is understood
• Define a discrete probability distribution to be the set of values $x$ taken by a discrete random variable $X$, together with the probabilities $P(x)$ that $x$ is the outcome of the experiment
• Define a discrete random variable to be uniformly distributed if it has finitely many values, all with the same probability, and use it to model random phenomena with equally likely outcomes
• Recognise the mean, or expected value, $\mu$ or $E(X)$, of a discrete random variable $X$ as a measure of centre for its distribution
• Generate and use the formulas $E(X)=\mu =\Sigma xP(x)$ and $Var(X)={\sigma }^2=\Sigma (x-\mu {)}^{2}P(x)=\Sigma x^{2}P(x)-{\mu }^2$ for the random variable $X$, where $Var(X)$ is the variance and $\sigma$ is the standard deviation of the distribution
• Recognise that the variance is an expected value because $Var(X)=E((X-\mu {)}^2)$
• Generate a probability distribution for a given discrete random variable and represent the probability distribution in graphical and tabular form
• Solve problems involving probabilities, expectation and variance of discrete random variables

Continuous random variables

• Estimate the probability that a continuous random variable falls in some interval using relative frequencies and histograms obtained from data
• Recognise that the probability of a particular value for a continuous random variable is 0 and hence that $P(a<X<b)=P(a\le X<b)=P(a<X\le b)=P(a\le X\le b)$ since $P(X=a)=P(X=b)=0$ when $X$ is a continuous random variable
• Define the cumulative distribution function (CDF), $F(x)$, as the probability of a random variable, $X$, having values less than or equal to $x$, so $F(x)=P(X\le x)$ and $P(a\le x\le b)=F(b)-F(a)$
• Recognise that the cumulative distribution function, $F(x)$, is non-decreasing for all $x$ in its domain, and graph cumulative distribution functions, given a formula for $F(x)$, with and without graphing applications
• Define a probability density function (PDF), $f(x)$, for a random variable $X$ with cumulative distribution function $F(x)$ as $f(x)=F^{\prime }(x)$ and recognise that $P(a<X<b)={\int }_a^{b}f(x)dx$
• Recognise the properties of a probability density function: $f(x)\ge 0$ for all $x$ in the domain of $f(x)$, and ${\int }_a^{b}f(x)dx=1$ if the domain of $f(x)$ is $[a,b]$, or ${\int }_{-\infty }^{\infty }f(x)dx=1$ if the domain of $f(x)$ is all real $x$
• Apply the properties of a probability density function to solve problems and justify conclusions
• Find the mode from a given probability density function
• Obtain the cumulative distribution function using the formula $F(x)={\int }_a^{x}f(t)dt$ where $f(x)$ is a given probability density function defined on the interval $[a,b]$
• Determine and use the probability density function $f(x)=\frac{1}{b-a}$ for a continuous uniform distribution for a random variable $X$ taking values in the interval $[a,b]$
• Use a cumulative distribution function to calculate the median and quartiles for a continuous random variable
• Find the probability density function from a given cumulative distribution function
• Generate the expression for the expected value of a continuous random variable, $E(X)=\mu ={\int }_a^{b}xf(x)dx$, where the probability density function $f(x)$ is defined on the interval $[a,b]$
• Generate the expression for the variance of a continuous random variable, $Var(X)=E(X^2)-{\mu }^2={\int }_a^b{x}^{2}f(x)dx-{\mu }^2$, where the probability density function $f(x)$ is defined on the interval $[a,b]$
• Evaluate the expected value and the variance of a continuous random variable, where the probability density function $f(x)$ is defined on the interval $[a,b]$, that involve integration of functions within the scope of the Mathematics Advanced course
• Evaluate the expected value and the variance of a continuous random variable, where the probability density function $f(x)$ is defined on the interval $[a,b]$, that involve integration of functions beyond the scope of the Mathematics Advanced course using an online computational application

The normal distribution

• Identify the normal distribution as a continuous probability distribution that is used to model many naturally occurring phenomena
• Identify the graph of the probability density function of a normal distribution, the normal curve, as an ‘ideal’ bell-shaped curve, symmetrical about its mean which is equal to its mode and median, and as having most values concentrated about the mean
• Identify contexts that can be approximately modelled by a normal random variable
• Use the notation $X\sim N(\mu ,{\sigma }^2)$ to represent a normally distributed random variable that has mean $\mu$ and standard deviation $\sigma$
• Represent probabilities associated with the normal distribution by areas of shaded regions under the normal curve, which may extend to $\pm \infty$
• Apply the empirical rule to make judgements and solve problems involving probabilities of normally distributed data: that for normal distributions, approximately 68% of data lie within one standard deviation of the mean, approximately 95% within two standard deviations of the mean and approximately 99.7% within three standard deviations of the mean
• Use graphing applications to explore the normal distribution, graph the probability density function $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$, verify the empirical rule and graph the cumulative distribution function
• Recognise features of the normal curve, and identify the global maximum and points of inflection
• Distinguish between a standard normal distribution with mean 0 and standard deviation 1, and the non-standard normal distribution with mean $\mu$ and standard deviation $\sigma$
• Define the $z$-score, or standardised score, by the formula $z=\frac{x-\mu }{\sigma }$, where $\mu$ is the mean and $\sigma$ is the standard deviation, and $x$ is an observed value of a random variable
• Interpret the $z$-score as the number of standard deviations a score lies above or below the mean
• Use $z$-scores to compare scores from different sets of data and justify conclusions
• Use $z$-scores to identify probabilities of events less or more extreme than a given outcome and solve problems using tables for the standard normal distribution
• Solve problems involving finding the mean or standard deviation of a normal random variable given the probability of an event less or more extreme than a given outcome
• Use $z$-scores to make judgements related to probabilities of certain events or given sets of data assuming an underlying normal distribution