6: Random Variables and Probability Distributions

Random Variables

Random variables are defined as a variable that depends on the outcomes of a chance experiment, which are numerical values

Types of Random Variables

Discrete

Variables which are measurable and countable

Continuous

Variables which can consist of any numeric value

Properties of Probability Distributions

Expected Values and Standard Deviations

\[ E=\Sigma x\times p(x)\\ \sigma_x=\Sigma(x-\mu_x)^2p(x) \]

Linear Combinations

Properties of Data Points

Let \(x\) be any data point, and \(c\) any constant

Addition and Subtraction

Represented by \(x+c\)

\[ \mu =\mu+c\\ \sigma=\sigma \]

Multiplication and Subtraction

Represented with \(xc\)

\[ \mu=c\mu\\ \sigma=c\sigma \]

Variances and Standard Deviations

\[ \sigma^2=\Sigma\sigma^2\\ \therefore\sigma=\sqrt{\Sigma\sigma^2} \]

Distributions

Binomial Distributions

Binomial distributions have two possibilities, most commonly true or false. \(P(\text{success})\) must be constant every round, trials must be independent, and there are a finite amount of trials

\[ \mu_x=np\\ \sigma_x=\sqrt{np(1-p)} \]

Geometric Distributions

Geometric distributions have two possibilities, most commonly true or false. \(P(\text{success})\) must be constant every round, trials must be independent, you keep going until first success. Geometric distributions must atleast have one trial

\[ P(x)=(1-P(s))^{n-1}P(s)\\ \mu_x=\frac{1}{p}\\ \sigma_x=\frac{\sqrt{1-p}}{p} \]

Normal Distributions

Combination of two normal distributions