```{r setup, include=FALSE} knitr::opts_chunk$set(echo = FALSE) ``` ```{r, echo=F, message=F, warning=F} library(readr) library(openintro) data(COL) ``` # Geometric distribution ## Milgram experiment \begin{multicols}{2} \small \begin{itemize} \item Stanley Milgram, a Yale University psychologist, conducted a series of experiments on obedience to authority starting in 1963. \item Experimenter(E) orders the teacher (T), the subject of the experiment, to give severe electric shocks to a learner (L) each time the learner answers a question incorrectly. \item The learner is actually an actor, and the electric shocks are not real, but a prerecorded sound is played each time the teacher administers an electric shock. \end{itemize} \normalsize \columnbreak \includegraphics[width=1\columnwidth]{milgram.png} \end{multicols} ## Milgram experiment - These experiments measured the willingness of study participants to obey an authority figure who instructed them to perform acts that conflicted with their personal conscience. - Milgram found that about 65% of people would obey authority and give such shocks. - Over the years, additional research suggested this number is approximately consistent across communities and time. ## Bernouilli random variables - Each person in Milgram's experiment can be thought of as a **trial**. - A person is labeled a **success** if she refuses to administer a severe shock, and **failure** if she administers such shock. - Since only 35% of people refused to administer a shock, **probability of success** is **p = 0.35**. - When an individual trial has only two possible outcomes, it is called a **Bernoulli random variable**. ## Geometric distribution \alert{Dr. Smith wants to repeat Milgram's experiments but she only wants to sample people until she finds someone who will not inflict a severe shock. What is the probability that she stops after the first person?} \centering{P($1^{st}$ person refuses) = 0.35} ## Geometric distribution \alert{Dr. Smith wants to repeat Milgram's experiments but she only wants to sample people until she finds someone who will not inflict a severe shock. What is the probability that she stops after the first person?} \centering{P($1^{st}$ person refuses) = 0.35} \raggedright \alert{... the third person?} \centering{P($1^{st}$ and $2^{nd}$ shock, $3^{rd}$ refuses) \\ = $\frac{S}{0.65} \times \frac{S}{0.65} \times \frac{R}{0.35} = 0.65^2 \times 0.35 \approx 0.15$} ## Geometric distribution \alert{Dr. Smith wants to repeat Milgram's experiments but she only wants to sample people until she finds someone who will not inflict a severe shock. What is the probability that she stops after the first person?} \centering{P($1^{st}$ person refuses) = 0.35} \raggedright \alert{... the third person?} \centering{P($1^{st}$ and $2^{nd}$ shock, $3^{rd}$ refuses) \\ = $\frac{S}{0.65} \times \frac{S}{0.65} \times \frac{R}{0.35} = 0.65^2 \times 0.35 \approx 0.15$} \raggedright \alert{... the tenth person?} ## Geometric distribution \alert{Dr. Smith wants to repeat Milgram's experiments but she only wants to sample people until she finds someone who will not inflict a severe shock. What is the probability that she stops after the first person?} \centering{P($1^{st}$ person refuses) = 0.35} \raggedright \alert{... the third person?} \centering{P($1^{st}$ and $2^{nd}$ shock, $3^{rd}$ refuses) \\ = $\frac{S}{0.65} \times \frac{S}{0.65} \times \frac{R}{0.35} = 0.65^2 \times 0.35 \approx 0.15$} \raggedright \alert{... the tenth person?} \centering{P($1$ shocks, $10^{th}$ refuses) \\ = $\underbrace{\frac{S}{0.65} \dots \frac{S}{0.65}} \times \frac{R}{0.35} = 0.65^9 \times 0.35 \approx 0.0072$} ## Geometric distribution **Geometric distribution** describes the waiting time until a success for **independent and identically distributed (iid)** Bernoulli random variables. - Independence: Outcomes of trials don't affect each other. - Identical: The probability of success is the same for each trial. ## Geometric distribution **Geometric distribution** describes the waiting time until a success for **independent and identically distributed (iid)** Bernoulli random variables. - Independence: Outcomes of trials don't affect each other. - Identical: The probability of success is the same for each trial. **Geometric probabilities** If $p$ represents probability of success, $(1-p)$ represents probability of failure, and $n$ represents number of independent trials \centering{P(success on the $n^{th}$ trial) = $(1-p)^{n-1}p$} ## Practice \alert{Can we calculate the probability of rolling a 6 for the first time on the $6^{th}$ roll of a die using the geometric distribution? Note that what was a success (rolling a 6) and what was a failure (not rolling a 6) are clearly defined and one or the other must happen for each trial.} A) No, on the the roll of a die there are more than 2 possible outcomes. B) Yes, why not? ## Practice \alert{Can we calculate the probability of rolling a 6 for the first time on the $6^{th}$ roll of a die using the geometric distribution? Note that what was a success (rolling a 6) and what was a failure (not rolling a 6) are clearly defined and one or the other must happen for each trial.} A) No, on the the roll of a die there are more than 2 possible outcomes. B) **Yes, why not?** \centering{P(6 on the $6^{th}$ roll) = $(\frac{5}{6})^5 \cdot (\frac{1}{6}) \approx 0.067$} ## Expected value \alert{How many people is Dr. Smith expected to test before finding the first one that refuses to administer the shock?} ## Expected value \alert{How many people is Dr. Smith expected to test before finding the first one that refuses to administer the shock?} The expected value, or the mean, of a geometric distribution is defined as $\frac{1}{p}$. \centering{$\mu = \frac{1}{p} = \frac{1}{0.35} = 2.86$} ## Expected value \alert{How many people is Dr. Smith expected to test before finding the first one that refuses to administer the shock?} The expected value, or the mean, of a geometric distribution is defined as $\frac{1}{p}$. \centering{$\mu = \frac{1}{p} = \frac{1}{0.35} = 2.86$} \raggedright She is expected to test 2.86 people before finding the first one that refuses to administer the shock. ## Expected value \alert{How many people is Dr. Smith expected to test before finding the first one that refuses to administer the shock?} The expected value, or the mean, of a geometric distribution is defined as $\frac{1}{p}$. \centering{$\mu = \frac{1}{p} = \frac{1}{0.35} = 2.86$} \raggedright She is expected to test 2.86 people before finding the first one that refuses to administer the shock. But how can she test a non-whole number of people? ## Expected value and its variability Mean and standard deviation of geometric distribution \centering{$\mu=\frac{1}{p}$ \hspace{1.5cm} $\sigma = \sqrt{\frac{1-p}{p^2}}$} ## Expected value and its variability Mean and standard deviation of geometric distribution \centering{$\mu=\frac{1}{p}$ \hspace{1.5cm} $\sigma = \sqrt{\frac{1-p}{p^2}}$} - Going back to Dr. Smith's experiment: \centering{$\sigma = \sqrt{\frac{1-p}{p^2}}=\sqrt{\frac{1-0.35}{0.35^2}}=2.3$} ## Expected value and its variability Mean and standard deviation of geometric distribution \centering{$\mu=\frac{1}{p}$ \hspace{1.5cm} $\sigma = \sqrt{\frac{1-p}{p^2}}$} - Going back to Dr. Smith's experiment: \centering{$\sigma = \sqrt{\frac{1-p}{p^2}}=\sqrt{\frac{1-0.35}{0.35^2}}=2.3$} - Dr. Smith is expected to test 2.86 people before finding the first one that refuses to administer the shock, give or take 2.3 people. ## Expected value and its variability Mean and standard deviation of geometric distribution \centering{$\mu=\frac{1}{p}$ \hspace{1.5cm} $\sigma = \sqrt{\frac{1-p}{p^2}}$} - Going back to Dr. Smith's experiment: \centering{$\sigma = \sqrt{\frac{1-p}{p^2}}=\sqrt{\frac{1-0.35}{0.35^2}}=2.3$} - Dr. Smith is expected to test 2.86 people before finding the first one that refuses to administer the shock, give or take 2.3 people. - These values only make sense in the context of repeating the experiment many many times.
