2.1 Review of Discrete random variables

This section will give a review of discrete random variables. All this information should be familiar to you from Math103 but is given again here for you to refer to easily during the course. If you need more description than provided here, please refer to your Math103 notes or the additional notes on Moodle.

Discrete random variables are simply random variables that are only defined for certain values of $r$ e.g. $0,1,\mathrm{\dots }$. Each possible value has a quantifiable probability of occurrence. Crucially the sum over all possible values should equal 1. The values and associated probabilities are usually shown in a probability distribution table.

Some important quantities of discrete random variables are:

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  Expected value of a discrete random variable = sum of the probability of each possible outcome times the outcome value (or payoff). $𝔼\left(R\right)=\mathbb{P}(R=r_1)\times r_1+\mathbb{P}(R=r_2)\times r_2+\mathrm{\dots }+\mathbb{P}(R=r_n)\times r_n={\sum }_{r}rp(r)$.
  Sample mean: $\overline{r}=\frac{1}{n}{\sum }_i{r}_i$.

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  Expected value of a function of the random variable = sum of the probability of each possible outcome times the function of the outcome value. $𝔼\left(g(R)\right)=\mathbb{P}(R=r_1)\times g(r_1)+\mathbb{P}(R=r_2)\times g(r_2)+\mathrm{\dots }+\mathbb{P}(R=r_n)\times g(r_n)={\sum }_{r}g(r)p(r)$.

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  Variance: $\text{Var}(R)={\sigma }^2=𝔼(R^2))-[𝔼\left(R\right)]{}^2$.
  Sample variance: $s_r^2=\frac{1}{n-1}{\sum }_{i=1}^n{(r_i-\overline{r})}^2$.

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  Standard deviation $\sigma (R)=\sqrt{\text{Variance}}$.
  Sample standard deviation: $s_r=\sqrt{s_r^2}$.

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  Density or Probability Mass Function (PMF): $p(r)=\mathbb{P}(R=r)$.

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  Cumulative Distribution Function (CDF): $F(r)=\mathbb{P}(R\le r)={\sum }_{i=0}^r\mathbb{P}(R=r)$. Note: the CDF always takes values between 0 and 1 and is an increasing function.