class: center, middle, inverse, title-slide # Binomial Distribution ### 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> --- ## Conditions - The trials have to be independent from each other. - The probability of success has to be the same for each trial. - The number of trials is fixed. - Each trial outcome is either a success or a failure. --- ## Binomial Distribution The random variable X represents the number of successes in `\(n\)` trials where in independent trial the probability of success is `\(p\)`. -- `\(X\sim \text{Binomial}(n, p)\)` -- `\(P(X = x) = f(x) = {n \choose x}p^{x} (1-p)^{n-x}\)` -- `\(S = \{0,1,2...,n\}\)` -- `\(E(X) = np\)` -- `\(Var(X) = np(1-p)\)` --- __Example__ A vet has been assigned to work at a farm where 70% of the animals have been infected by avian influenza. The vet selects 10 random animals to inspect. What is the probability that 3 of the selected animals are infected? -- `\(n = 10\)`, `\(x = 3\)`, `\(p = 0.70\)` -- `\(P(X = 3) = f(3) = {10 \choose 3}0.70^{3} (1-0.70)^{10-3}\)` -- `\(P(X = 3)= \frac{10!}{3!7!}0.70^30.30^7=0.009001692\)` -- ```r dbinom(x = 3, size = 10, prob = 0.70) ``` ``` ## [1] 0.009001692 ``` --- class: center middle ## pmf when `\(n = 10\)` and `\(p = 0.70\)` <img src="slide-2-binomial_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" /> --- ## Cumulative Probability Function What is the probability that at most three of the ten selected animals are infected? `\(P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)\)` -- `\(\frac{10!}{10!0!}0.70^{0}0.30^10 + \frac{10!}{9!1!}0.70^{1}0.30^9 + \frac{10!}{8!2!}0.70^{2}0.30^8 + \frac{10!}{7!3!}0.70^{3}0.30^7\)` -- ```r dbinom(x = 0, size = 10, prob = 0.70) + dbinom(x = 1, size = 10, prob = 0.70) + dbinom(x = 2, size = 10, prob = 0.70) + dbinom(x = 3, size = 10, prob = 0.70) ``` ``` ## [1] 0.01059208 ``` --- ## Cumulative Probability Function `\(P(X \leq 3)\)` ```r pbinom(q = 3, size = 10, prob = 0.70) ``` ``` ## [1] 0.01059208 ``` --- ## Expected Value What is the expected value of number of infected animals in 10 selected animals? -- `\(E(X) = np\)` `\(E(X) = 10\times0.70=7\)` -- ## Variance What is the variance of number of infected animals in 10 selected animals? -- `\(Var(X) = np(1-p)\)` `\(Var(X) = 10\times0.7(1-0.7) = 2.1\)`