class: center, middle, inverse, title-slide # Discrete Random Variables Review ### Dr. Dogucu --- layout: true <div class="my-header"></div> <div class="my-footer"> Copyright &copy; <a href="https://mdogucu.ics.uci.edu">Dr. Mine Dogucu</a>. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div> --- A **discrete random** variable has a countable number of possible numeric outcomes. -- Probability mass function (pmf): `\(P(X = x) = f(x)\)` -- Cumulative distribution function (cdf): `\(P(X \leq x) = F(x)\)` -- `\(E(X) = \sum_{S} x f(x)\)` -- `\(Var(X)= E(X^2) - [E(X)]^2\)` --- ## Bernoulli Distribution If X is a random variable that takes value 1 with probability of success `\(p\)` and 0 with probability `\(1-p\)`, then X follows a Bernoulli distribution. `\(S = \{0, 1\}\)` `\(X \sim \text{Bernoulli} (p)\)` -- `\(E(X) = \mu = p\)` -- `\(Var(X)=\sigma^2 = p(1-p)\)` --- ## Geometric Distribution Let X be the number of failures needed __before__ the first success is observed in independent trials. `\(X\)` follows a geometric distribution .pull-left[ `\(S = \{0, 1, 2, 3, 4, ...\}\)` `\(X \sim \text{Geometric} (p)\)` `\(f(x) = (1-p)^x(p)\)` `\(E(X)=\frac{1-p}{p}\)` `\(Var(X) = \frac{1-p}{p^2}\)` ] .pull-right[ ```r dgeom(x, prob) #pmf ``` ```r pgeom(q, prob) #cdf ``` ] --- ## Binomial Distribution The random variable X represents the number of successes in `\(n\)` trials where in independent trial the probability of success is `\(p\)`. .pull-left[ `\(S = \{0, 1, ..., n \}\)` `\(X\sim \text{Binomial}(n, p)\)` `\(P(X = x) = f(x) = {n \choose x}p^{x} (1-p)^{n-x}\)` `\(E(X) = np\)` `\(Var(X) = np(1-p)\)` ] .pull-right[ ```r dbinom(x, size, prob) #pmf ``` ```r pbinom(q, size, prob) #cdf ``` ] --- ## Poisson Distribution Let `\(X\)` represent the number of occurrences of an event within a __fixed__ time or space. .pull-left[ `\(S = \{0, 1, 2, 3, 4, ...\}\)` `\(X \sim Poisson (\lambda)\)` `\(P(X = x) = f(x) =\frac{\lambda^x}{x!} e^{-\lambda}\)` `\(E(X) = Var(X) = \lambda\)` ] .pull-right[ ```r dpois(x, lambda) #pmf ``` ```r ppois(q, lambda) #cdf ``` ]