Announcements : Midterm 2

• Midterm next Friday 11/15

  • Will cover material from after Midterm 1 until today's lecture
  • Monday: holiday
  • Wednesday: review (OR)
  • Thursday: no lab, XHT OH 12-1pm on Zoom
  • Same rules apply: closed-book, no computers or calculators, no make-up exams

• These formulas will be provided:

  • Bayes' theorem: $P(A \mid B) =\frac{P(B \mid A)P(A)}{P(B)}$.
  • Probability mass functions:
    • Binomial: $P(X=x)=\begin{pmatrix} n \\ x \end{pmatrix}p^x(1-p)^{n-x}$
    • Poisson: $P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}$, $\lambda > 0$

Today

• Common probability distributions

  • Poisson distribution

Random samples from binomial distribution

• Use rbinom() to get 5000 draws from the population

• In R:

  set.seed(0) # so results are reproducible 
  binomDraws <- rbinom(n = 5000, size = 3, prob = .2)
  table(binomDraws)/5000

  ## binomDraws
  ##      0      1      2      3 
  ## 0.5246 0.3638 0.1040 0.0076

Frequency distribution for Binomial(3, .2)

data.frame(binomDraws) %>%
  ggplot(aes(x = binomDraws)) + 
    geom_bar() +
    labs(x = "Number of Smokers",
         title = "5000 samples from Binomial(3, .2)")

Varying the number of Bernoulli trials: 100 trials

set.seed(0) # so results are reproducible 
binomDraws100 <- rbinom(n = 5000, size = 100, prob = .2)
data.frame(binomDraws100) %>%
  ggplot(aes(x = binomDraws100)) + 
    geom_bar() +
    labs(x = "Number of Smokers",
         title = "5000 samples from Binomial(100, .2)")

Varying the number of Bernoulli trials: 500 trials

set.seed(0) # so results are reproducible 
binomDraws500 <- rbinom(n = 5000, size = 500, prob = .2)
data.frame(binomDraws500) %>%
  ggplot(aes(x = binomDraws500)) + 
    geom_bar() +
    labs(x = "Number of Smokers",
         title = "5000 samples from Binomial(500, .2)")

Frequency distribution varying number of Bernoulli trials

data.frame(binomDraws) %>%
  bind_cols(size = 3) %>%
  bind_rows(
    data.frame(binomDraws100) %>%
      rename(binomDraws = binomDraws100) %>%
      bind_cols(size = 100)
  ) %>%
  bind_rows(
    data.frame(binomDraws500) %>%
      rename(binomDraws = binomDraws500) %>%
      bind_cols(size = 500)
  ) %>%
  ggplot(aes(x = binomDraws, 
             fill = as.factor(size))) +
    geom_histogram(binwidth = 1, position = "identity", alpha = .7) + 
    labs(
      x = "Number of smokers",
      y = "Frequency",
      title = "5000 samples each from Binomial(3, .2), Binomial(100, .2), Binomial(500, .2)",
      fill = "Size"
    )

Frequency distribution varying probability of success

set.seed(0) # so results are reproducible 
binomP.2 <- rbinom(n = 5000, size = 100, prob = .2)
binomP.5 <- rbinom(n = 5000, size = 100, prob = .5)
binomP.7 <- rbinom(n = 5000, size = 100, prob = .7)

Poisson distribution

• Useful for estimating the number of events in a large population over a unit of time.

• For example:

  • The number of people having heart attacks in New York City every year
  • The number of accidents occurring at an intersection per hour
  • The number of typos in every 100 pages of a book

• It is named after French mathematician Siméon Denis Poisson

Poisson distribution

• E.g.: Number of people having heart attacks in New York City every year

• Key ingredients

  • Fixed interval of time or space

  • Events happen with a known average rate, independently of time since the last event ("memoryless" property)

    • One person having a heart attack does not change the probability of another person having a heart attack, hence the timing of the next heart attack

• The parameter that defines a Poisson distributed random variable is the rate $\lambda$, where $\lambda > 0$

  • Rate = average number of occurrences per unit of time

• Often used to model rare events

Probability mass function, mean and variance

• $P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}$, defined over non-negative integer values of $x$

  • Recall: $n! = n(n - 1)(n - 2)\cdots (1)$.

• No upper limit, i.e., $x$ can take very large non-negative integer values

• $E(X) = \lambda$

• $Var(X) = \lambda$

Poisson probabilities

• $P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}$ lets us calculate probabilities of taking a certain value

• For $x = 2$ and $\lambda = 3$, we have

$$\begin{aligned} P(X = 2) &= \frac{3^2 e^{-3}}{2!} = \frac{9(e^{-3})}{2(1)} = 0.2240418 \end{aligned}$$

• In R:

dpois(x = 2, lambda = 3)

## [1] 0.2240418

• For large values of $x$, the probability is very small because of the large denominator

dpois(x = 10, lambda = 3)

## [1] 0.0008101512

Probability distribution

• In the same manner, we can derive the entire probability distribution

dpois(x = 0:10, lambda = 3)

##  [1] 0.0497870684 0.1493612051 0.2240418077 0.2240418077
##  [5] 0.1680313557 0.1008188134 0.0504094067 0.0216040315
##  [9] 0.0081015118 0.0027005039 0.0008101512

data.frame(x = 0:10, y = dpois(0:10, lambda = 3)) %>%
  ggplot(aes(x = x, y = y)) + 
    geom_bar(stat = "identity") +
  labs(title = "Probability distribution of Poisson(3)",
       y = "P(X = x)")

Probability distribution varying lambda

data.frame(x = 0:30, y = dpois(0:30, lambda = 3), lambda = 3) %>%
  bind_rows(data.frame(x = 0:30, y = dpois(0:30, lambda = 10), lambda = 10)) %>%
  bind_rows(data.frame(x = 0:30, y = dpois(0:30, lambda = 20), lambda = 20)) %>%
    ggplot(aes(x = x, y = y, fill = as.factor(lambda))) + 
      geom_bar(stat = "identity", 
               position = "identity", 
               alpha = .5) +
    labs(title = "Probability distribution of \nPoisson(3), Poisson(10), Poisson(20)",
         y = "P(X = x)",
         fill = "Lambda")

Sampling from Poisson distribution in R

• Simulate random draws using the rpois() function

• rpois() has the arguments

  • n, the number of draws from the distribution
  • lambda, the mean

set.seed(0) # so results are reproducible 
inputLambda <- 3
poissonDraws <- rpois(n = 100, lambda = inputLambda)
poissonDraws

##   [1] 5 2 2 3 5 2 5 6 4 3 1 2 1 4 2 4 3 4 8 2 4 6 2 4 1 2 2 0 2 5
##  [31] 2 3 3 3 1 5 4 4 1 4 2 5 3 4 3 3 4 0 3 4 4 3 5 3 2 1 1 2 3 4
##  [61] 2 5 2 3 2 4 2 3 4 1 5 2 5 2 2 3 5 5 2 4 6 3 4 2 2 4 2 4 1 2
##  [91] 1 2 1 3 5 4 4 3 2 4

Frequency distribution varying lambda

set.seed(0) # so results are reproducible 
poissonL3 <- rpois(n = 5000, lambda = 3)
poissonL10 <- rpois(n = 5000, lambda = 10)
poissonL20 <- rpois(n = 5000, lambda = 20)

data.frame(poissonL3) %>%
  rename(outcome = poissonL3) %>%
  bind_cols(lambda = 3) %>%
  bind_rows(
    data.frame(poissonL10) %>%
      rename(outcome = poissonL10) %>%
      bind_cols(lambda = 10)
  ) %>%
  bind_rows(
    data.frame(poissonL20) %>%
      rename(outcome = poissonL20) %>%
      bind_cols(lambda = 20)
  ) %>%
  ggplot(aes(x = outcome, 
                    fill = as.factor(lambda))) +
    geom_histogram(binwidth = 1, position = "identity", alpha = .7) + 
    labs(
      x = "Number of occurrences",
      y = "Frequency",
      title = "5000 samples each from \nPoisson(3), Poisson(10), Poisson(20)",
      fill = "Lambda"
    )

Exercises

An insurance agency determines that 70% of individuals do not exceed their deductible.

• Suppose the insurance agency is considering a random sample of four individuals they insure. What is the probability that exactly one of them will exceed the deductible?

• What is the probability that 3 of 8 randomly selected individuals will have exceeded the insurance deductible, i.e., that 5 of 8 will not exceed the deductible?

Exercises

A very skilled court stenographer makes one typographical error (typo) per hour on average.

• What probability distribution is most appropriate for calculating the probability of a given number of typos this stenographer makes in an hour?

• What are the mean and the standard deviation of the number of typos this stenographer makes?

• Would it be considered unusual if this stenographer made 4 or more typos in a given hour?

• Calculate the probability that this stenographer makes at most 2 typos in a given hour.

Summary

• Common probability distributions: Poisson

  • Theoretical properties: probability density function, parameters, mean and variance, effect of varying parameters

  • R functions, e.g.:

    • dpois() for densities
    • ppois() for $P(X\leq x)$
    • rpois() for random sample