Lesson 15: Cadet-Led Review

Calendar

Block I calendar with a red box around Friday 25 September, Lesson 15, Cadet-Led Review.

WPR I Admin

ImportantWPR I: Lesson 16 (29, 30 Sep)
  • Covers: Lessons 1 through 13
  • Time: 55 minutes
  • Authorized: the course statistics reference card (SRC), R-Lite, and the issued calculator

R-Lite and the SRC

WarningHave R-Lite working and know your SRC before you walk in
  • Download R-Lite from Canvas.
  • Open it and run one call, like pnorm(1), today.
  • Not working? Tell me today, not at the start of the WPR.
  • Open the SRC and know where every formula and table lives.

Study Materials

What We’re Doing: Lesson 15

Objectives

  • Review Lessons 1-13.

Required Reading

None

Break!

Family

Cadet-Led Review

Block I Outline

Descriptive Statistics

Lesson 1: Types of Data & Study Design (Devore 1.1, 1.2, S1)
  • Population, sample, and process
  • Descriptive vs inferential statistics
  • Parameter (\(\mu\), \(\sigma\), \(p\)) vs statistic (\(\bar{x}\), \(s\), \(\hat{p}\))
  • Categorical (nominal, ordinal) vs numerical (discrete, continuous)
  • Histograms and shape: symmetric, skewed left, skewed right (named for the tail)
  • Simple random sample
  • Observational study vs experiment
  • Random sampling buys generalization, random assignment buys causation
Lesson 2: Measures of Location & Variability (Devore 1.3, 1.4, S2)
  • Center: mean \(\bar{x}\), median \(\tilde{x}\), trimmed mean
  • Mean is sensitive to outliers, median is resistant
  • Spread: range, sample variance \(s^2 = \dfrac{\sum (x_i - \bar{x})^2}{n - 1}\), standard deviation \(s\)
  • Fourth spread \(f_s\) = upper fourth minus lower fourth
  • Outlier: more than \(1.5 f_s\) beyond the nearest fourth, extreme beyond \(3 f_s\)
  • Boxplots and comparative boxplots

Probability

Lesson 3: Set Theory (Devore 2.1)
  • Experiment, sample space \(\mathcal{S}\), event (a subset of \(\mathcal{S}\))
  • Union \(A \cup B\) (“or”), intersection \(A \cap B\) (“and”), complement \(A'\) (“not”)
  • Mutually exclusive (disjoint): \(A \cap B = \emptyset\)
  • Venn diagrams
  • DeMorgan’s laws: \((A \cup B)' = A' \cap B'\) and \((A \cap B)' = A' \cup B'\)
Lesson 4: Probability Basics (Devore 2.2)
  • Axioms: \(P(A) \ge 0\), \(P(\mathcal{S}) = 1\), and disjoint probabilities add
  • Complement rule: \(P(A') = 1 - P(A)\)
  • Addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), and its three event version
  • Equally likely outcomes: \(P(A) = N(A)/N\)
  • Two way tables
Lesson 5: Counting (Devore 2.3)
  • Product rule: \(n_1 n_2 \cdots n_k\)
  • Permutation (order matters): \(P_{k,n} = \dfrac{n!}{(n-k)!}\)
  • Combination (order does not): \(\dbinom{n}{k} = \dfrac{n!}{k!\,(n-k)!}\)
  • Counting to get probabilities: \(P(A) = N(A)/N\)
Lesson 6: Conditional Probability (Devore 2.4)
  • Conditional probability: \(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)
  • Multiplication rule: \(P(A \cap B) = P(A \mid B)\,P(B)\)
  • Tree diagrams
  • Law of Total Probability: \(P(B) = \sum_i P(B \mid A_i)\,P(A_i)\)
  • Bayes’ Theorem: \(P(A_j \mid B) = \dfrac{P(B \mid A_j)\,P(A_j)}{\sum_i P(B \mid A_i)\,P(A_i)}\)
Lesson 7: Independence (Devore 2.5)
  • Independent: \(P(A \mid B) = P(A)\)
  • Test with \(P(A \cap B) = P(A)\,P(B)\)
  • Independent is not the same as mutually exclusive
  • \(P(\text{at least one}) = 1 - P(\text{none})\)
  • Counting combined with independence

Random Variables

Lesson 8: Discrete Random Variables (Devore 3.1, 3.2, 3.3)
  • Random variable: a number assigned to each outcome
  • pmf: \(p(x) = P(X = x)\), with \(p(x) \ge 0\) and \(\sum_x p(x) = 1\)
  • cdf: \(F(x) = P(X \le x)\), a step function
  • Expected value: \(E(X) = \mu = \sum_x x\,p(x)\)
  • Variance: \(V(X) = \sigma^2 = E(X^2) - [E(X)]^2\), and \(\sigma = \sqrt{V(X)}\)
Lesson 9: Binomial Distribution (Devore 3.4)
  • BINS: binary outcomes, independent trials, fixed \(n\), same \(p\)
  • \(X \sim \text{Bin}(n, p)\) counts successes
  • pmf: \(p(x) = \dbinom{n}{x} p^x (1-p)^{n-x}\) for \(x = 0, 1, \ldots, n\)
  • \(E(X) = np\) and \(V(X) = np(1-p)\)
  • R-Lite: dbinom is the pmf, pbinom is the cdf
  • Fully specify: name the variable, name the distribution, give the parameters
Lesson 10: Poisson Distribution (Devore 3.6)
  • Counts events over a window: time, distance, or area
  • Poisson process: rate \(\lambda\), window \(t\), so \(\mu = \lambda t\)
  • pmf: \(p(x; \mu) = \dfrac{e^{-\mu}\mu^x}{x!}\) for \(x = 0, 1, 2, \ldots\)
  • \(E(X) = V(X) = \mu\)
  • R-Lite: dpois is the pmf, ppois is the cdf
Lesson 11: Continuous Random Variables (Devore 4.1, 4.2)
  • pdf: \(f(x) \ge 0\) and \(\int_{-\infty}^{\infty} f(x)\,dx = 1\)
  • Probability is area: \(P(a \le X \le b) = \int_a^b f(x)\,dx\)
  • \(P(X = c) = 0\), so \(\le\) and \(<\) give the same answer
  • cdf: \(F(x) = P(X \le x) = \int_{-\infty}^{x} f(y)\,dy\), and \(P(a \le X \le b) = F(b) - F(a)\)
  • Percentile: solve \(F(c) = P(X \le c) = p\) for \(c\)
  • \(E(X) = \int x\,f(x)\,dx\) and \(V(X) = E(X^2) - [E(X)]^2\)
Lesson 12: Normal Distribution (Devore 4.3)
  • \(X \sim N(\mu, \sigma^2)\): symmetric, bell shaped, centered at \(\mu\)
  • Standard normal: \(Z \sim N(0, 1)\)
  • Standardize: \(z = \dfrac{x - \mu}{\sigma}\), then use the \(z\) table
  • Percentiles: \(x = \mu + z\sigma\)
  • Critical value: \(z_\alpha\) has area \(\alpha\) to its right
  • Empirical rule: 68%, 95%, 99.7% within 1, 2, 3 standard deviations
  • R-Lite: pnorm(x, mean, sd) is the cdf, qnorm runs it backwards
Lesson 13: Exponential Distribution (Devore 4.4)
  • \(X \sim \text{Exp}(\lambda)\) models a wait
  • pdf: \(f(x) = \lambda e^{-\lambda x}\) for \(x \ge 0\)
  • cdf: \(F(x) = P(X \le x) = 1 - e^{-\lambda x}\), so \(P(X > x) = e^{-\lambda x}\)
  • \(E(X) = \sigma = 1/\lambda\)
  • Memoryless: \(P(X \ge t + t_0 \mid X \ge t_0) = P(X \ge t)\)
  • Poisson counts events at rate \(\lambda\), the waits between them are \(\text{Exp}(\lambda)\)
  • R-Lite: pexp(x, rate) takes the rate, not the mean

Distributions

Binomial Poisson Normal Exponential
Type Discrete Discrete Continuous Continuous
Use it when \(x\) successes in \(n\) fixed trials \(x\) events in a window Symmetric, bell shaped measurement Wait until the next event
Fully specified \(X \sim \text{Binom}(n,\ p)\) \(X \sim \text{Pois}(\mu)\) \(X \sim N(\mu,\ \sigma^2)\) \(X \sim \text{Exp}(\lambda)\)
Parameters \(n\) trials, \(p\) success probability \(\mu = \lambda t\), expected count \(\mu\) center, \(\sigma\) spread \(\lambda\) rate
Possible values \(x = 0, 1, \dots, n\) \(x = 0, 1, 2, \dots\) \(-\infty < x < \infty\) \(x \ge 0\)
pmf or pdf \(\dbinom{n}{x} p^x (1-p)^{n-x}\) \(\dfrac{e^{-\mu}\mu^x}{x!}\) \(\dfrac{1}{\sqrt{2\pi}\,\sigma} e^{-(x-\mu)^2/(2\sigma^2)}\) \(\lambda e^{-\lambda x}\)
\(E(X)\) \(np\) \(\mu\) \(\mu\) \(1/\lambda\)
\(V(X)\) \(np(1-p)\) \(\mu\) \(\sigma^2\) \(1/\lambda^2\)
d, the pmf or pdf dbinom(x, n, p) dpois(x, mu) dnorm(x, mean, sd) dexp(x, rate)
p, the cdf \(P(X \le x)\) pbinom(x, n, p) ppois(x, mu) pnorm(x, mean, sd) pexp(x, rate)
q, the cdf backwards qbinom(area, n, p) qpois(area, mu) qnorm(area, mean, sd) qexp(area, rate)

Problem 1: Saturday Run

The distance \(X\) (in miles) a cadet runs on a Saturday morning has pdf

\[f(x) = \begin{cases} \dfrac{x}{24} & 1 \le x \le 7 \\ 0 & \text{otherwise} \end{cases}\]

  1. Find the cdf \(F(x) = P(X \le x)\) and write it in full piecewise form.
  2. Find \(P(X < 3)\).
  3. Find \(P(X \le 3)\).
  4. Find \(P(X > 6)\).
  5. Find \(P(X \ge 6)\).
  6. Find \(P(X = 6)\).
  7. Find \(\mu\) and \(\sigma\). Then find the probability \(X\) is within one standard deviation of its mean.
  1. For \(1 \le x \le 7\), \(\displaystyle\int_1^x \frac{y}{24}\,dy = \frac{x^2 - 1}{48}\), so

\[F(x) = P(X \le x) = \begin{cases} 0 & x < 1 \\ \dfrac{x^2 - 1}{48} & 1 \le x \le 7 \\ 1 & x > 7 \end{cases}\]

  1. \(F(3) = \dfrac{9 - 1}{48} = \dfrac{1}{6} \approx \mathbf{0.167}\)

  2. \(F(3) \approx \mathbf{0.167}\), the same as (b), because \(P(X = 3) = 0\).

  3. \(1 - F(6) = 1 - \dfrac{36 - 1}{48} = \dfrac{13}{48} \approx \mathbf{0.271}\)

  4. \(1 - F(6) \approx \mathbf{0.271}\), the same as (d).

  5. \(\mathbf{0}\). A single value has no area under the pdf. That is why (d) and (e) are equal.

  6. \(\mu = \displaystyle\int_1^7 x \cdot \frac{x}{24}\,dx = \frac{343 - 1}{72} = 4.75\) and \(E(X^2) = \displaystyle\int_1^7 x^2 \cdot \frac{x}{24}\,dx = \frac{2401 - 1}{96} = 25\), so \(V(X) = 25 - 4.75^2 = 2.4375\) and \(\sigma \approx 1.561\).

\(\mu \pm \sigma \approx (3.189,\ 6.311)\), so \(P(3.189 < X < 6.311) = F(6.311) - F(3.189) = \dfrac{6.311^2 - 3.189^2}{48} \approx \mathbf{0.618}\)

Problem 2: Motor Pool Work Orders

Work orders arrive at a battalion motor pool randomly and independently at an average rate of 5 per hour. Let \(X\) be the number of work orders in 1 hour.

  1. Fully specify the distribution of \(X\).
  2. Find \(P(X < 3)\).
  3. Find \(P(X \le 3)\).
  4. Find \(P(X > 7)\).
  5. Find \(P(X \ge 7)\).
  6. Find \(\mu\) and \(\sigma\). Then find the probability \(X\) is within one standard deviation of its mean.

  1. \(X \sim \text{Pois}(\mu = 5)\) work orders per hour.

  2. \(P(X < 3) = P(X \le 2) = F(2) \approx \mathbf{0.125}\) ppois(2, 5)

  3. \(F(3) \approx \mathbf{0.265}\) ppois(3, 5)

  4. \(1 - P(X \le 7) = 1 - F(7) \approx \mathbf{0.133}\) 1 - ppois(7, 5)

  5. \(1 - P(X \le 6) = 1 - F(6) \approx \mathbf{0.238}\) 1 - ppois(6, 5)

  6. \(\mu = 5\) and \(\sigma = \sqrt{5} \approx 2.236\). \(\mu \pm \sigma \approx (2.764,\ 7.236)\), so \(P(3 \le X \le 7) = F(7) - F(2) \approx \mathbf{0.742}\) ppois(7, 5) - ppois(2, 5)

Problem 3: Motor Pool Work Orders, Continued

Same motor pool, work orders still arrive at 5 per hour. Now let \(T\) be the time (in hours) between one work order and the next.

  1. Fully specify the distribution of \(T\). Write its pdf. Then find the cdf \(F(t) = P(T \le t)\) and write it in full piecewise form.
  2. Find \(P(T < 0.2)\).
  3. Find \(P(T \le 0.2)\).
  4. Find \(P(T > 0.5)\).
  5. Find \(P(T \ge 0.5)\).
  6. Find \(P(T = 0.5)\).
  7. Find \(\mu\) and \(\sigma\). Then find the probability \(T\) is more than two standard deviations above its mean.
  8. Find the 90th percentile of \(T\).
  1. The wait between Poisson events is exponential, so \(T \sim \text{Exp}(\lambda = 5)\) with \(f(t) = 5e^{-5t}\) for \(t \ge 0\). For \(t \ge 0\), \(\displaystyle\int_0^t 5e^{-5s}\,ds = 1 - e^{-5t}\), so

\[F(t) = P(T \le t) = \begin{cases} 0 & t < 0 \\ 1 - e^{-5t} & t \ge 0 \end{cases}\]

  1. \(F(0.2) = 1 - e^{-1} \approx \mathbf{0.632}\) pexp(0.2, 5)

  2. \(F(0.2) \approx \mathbf{0.632}\), the same as (b).

  3. \(1 - F(0.5) = e^{-2.5} \approx \mathbf{0.082}\) 1 - pexp(0.5, 5)

  4. \(1 - F(0.5) \approx \mathbf{0.082}\), the same as (d).

  5. \(\mathbf{0}\). A single value has no area under the pdf. That is why (d) and (e) are equal.

  6. \(\mu = \sigma = 1/\lambda = 0.2\) hours. \(\mu + 2\sigma = 0.6\), so \(P(T > 0.6) = 1 - F(0.6) = e^{-3} \approx \mathbf{0.050}\) 1 - pexp(0.6, 5)

  7. Solve \(F(c) = P(T \le c) = 0.90\) for \(c\):

\[1 - e^{-5c} = 0.90 \quad\Rightarrow\quad e^{-5c} = 0.10 \quad\Rightarrow\quad c = \frac{\ln(10)}{5} \approx \mathbf{0.461} \text{ hours}\]

qexp(0.90, 5)

Before You Leave

Today

  • Review Lessons 1-13

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Lesson 16: WPR I

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