Objectives for Exam 1

Introduction

This document outlines the objectives for Exam 1, covering the essential concepts from chapters 1 through 6.

Chapter 1: Introduction to Statistics

• Define and demonstrate knowledge of the three branches of statistics:

  • Data Collection: The process of gathering information.
  • Descriptive Statistics: Summarizing and organizing data.
  • Inferential Statistics: Drawing conclusions from data.

• Define and distinguish between a population and a sample including their respective symbols; population parameters by Greek letters, sample statistics are denoted by Latin letters.

• Determine whether a listing of objects refers to a population or a sample.

• Identify situations that exemplify probability or inferential statistics.

Chapter 2: Data Types and Distribution Shapes

• Identify data as univariate, bivariate, or multivariate.

• Recognize and classify variables as categorical/qualitative or numerical/quantitative.

• Describe the shape of a distribution:

  • Peaks: unimodal, bimodal, multimodal.
  • Symmetry: symmetric, right skewed, or left skewed.
  • Outliers: Identify and distinguish “real” outliers from the explicit points.

• Interpret histograms to describe shape and identification of outliers.

Chapter 3: Descriptive Statistics in R

• Given R output, identify the statistics: mean, median, variance, standard deviation, and quartiles.

• Understand and state the formulas for sample mean and sample variance:

\[\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\]

\[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2\]

• Calculate standard deviation from variance:

\[\text{Standard deviation} = \sqrt{\text{variance}}\]

• Calculate the Interquartile Range (IQR) and explain quartiles in non-mathematical terms. \[\text{IQR}=Q_3-Q_1\]

• Write down the five-number summary from R output and interpret modified boxplots.

• Using the five number summary identify inner and outer fences. \[\text{IF}_{\text{L}}=Q_1-1.5\times\text{IQR},~~\text{IF}_{\text{H}}=Q_3+1.5\times\text{IQR}\] \[\text{OF}_{\text{L}}=Q_1-3\times\text{IQR},~~\text{OF}_{\text{H}}=Q_3+3\times\text{IQR}\]

• Identify explicit points using the 1.5 IQR rule and evaluate if they are “real”.

• Draw/complete a modified boxplot from the five number summary and 1.5 IQR rule.

• Interpret the results of a modified boxplot or side-by-side boxplots.

• Decide on the appropriate measures of location and spread for given data.

Chapter 4: Probability

• Write down the sample space for experiments and determine disjoint events.

• Understand the frequentist interpretation of probability: \[\lim_{n \to \infty} \frac{n(E)}{n} \approx P(E)\]

• State and check the axioms associated with a probability space \(\Omega\): \[\text{For any event E}\subseteq\Omega,~~~~ 0 \leq P(E) \leq 1\] \[P(\Omega) = 1\] \[\text{For any event E}\subseteq\Omega,P(E)=\sum_{\omega\in \text{E}}P(\omega)\]

• Calculate a probability using:

  • The theoretical (or classical) approach (if equally likely can be assumed): \[P(A) = \frac{\text{Number of outcomes in A}}{\text{Total number of outcomes}}\]
  • Empirical approach using a provided probability distribution table.

• Use Venn diagrams to visualize and calculate probabilities.

• Calculate probabilities using probability rules:

  • General Addition Rule for any events \(A\) and \(B\) \[P(A \cup B) = P(A) + P(B)-P(A\cap B)\]
  • Special Addition Rule for disjoint events \(A\) and \(B\) \[P(A \cup B) = P(A) + P(B)\]
  • Complement Rule \[P(A') = 1 - P(A)\]
  • Law of Partitions: If \(B_1, ..., B_n\) are exhaustive and mutually exclusive events then \[P(A) = \sum_{i=1}^{n} P(A \cap B_i)\]
  • Law of Total Probability: If \(B_1, ..., B_n\) are exhaustive and mutually exclusive then \[P(A) = \sum_{i=1}^{n} P(A | B_i)P(B_i)\]
  • Calculate conditional probabilities using probability rules: \[P(B|A) = \frac{P(A \cap B)}{P(A)}\]
  • State and use the general multiplication rule to determine probabilities of intersections.
    • General Multiplication Rule for Two Events: \[P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)\]
    • General Multiplication Rule for Three Events: \[P(A \cap B \cap C) = P(A)P(B|A)P(C|A \cap B)=P(B)P(A|B)P(C|A\cap B)=P(C)P(B|C)P(A|B \cap C)=...\]

• Independence

  • Two events, \(A\) and \(B\), are independent if the occurrence of one does not affect the probability of the other: \(P(A | B) = P(A), ~~ (B | A) = P(B)\)
  • Special multiplication rule for independent events (Only use if you know for a fact they are independent.) \[P(A\cap B) = P(A) \times P(B)\]

• Bayes’ Rule:

• Baye’s Rule for 2 Events: \[P(A|B) = \frac{P(B|A)P(A)}{P(B | A)P(A)+P(B | A')P(A')}\]

• General Baye’s Rule for \(n\) Events: If \(A_1, ..., A_n\) are exhaustive and mutually exclusive events \[P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_{i=1}^{n} P(B | A_i)P(A_i)}\]

Chapter 5: Discrete Random Variables

• Recognize the properties of a valid probability distribution for discrete variables:

  • Each probability \[p_X(x)\] satisfies \[0 \leq p_X(x) \leq 1\].
  • The sum of all probabilities \(\sum p_X(x) = 1\).

• Calculate probabilities using a probability mass function (pmf).

• Calculate the mean of a discrete random variable (Expected value): \[\text{E}(X) = \mu_X = \sum x \cdot p(x)\]

• Calculate the variance and standard deviation for a discrete random variable:

  • Variance: \[\text{Var}(X) = \sigma_X^2 = \text{E}[(X - \mu_X)^2] = \text{E}(X^2) - [\text{E}(X)]^2\]

  • Standard deviation: \[\sigma_X = \sqrt{\text{Var}(X)}\]

Rules for Expected Value and Variance

• LOTUS For any real valued function \(g(\cdot)\) and discrete random variable \(X\) \[\text{E}[g(X)]=\sum_x g(x)p_X(x)\]

• Linearity of Expectation: For any two random variables \(X\) and \(Y\), and constants \(a\) and \(b\), \[ \text{E}(aX \pm bY) = a\text{E}(X) \pm b\text{E}(Y) \]

• Variance of a Linear function: For any random variable \(X\) and constants \(a\neq 0\) and \(b\), \[ \text{Var}(aX + b) = a^2\text{Var}(X) \] This shows that adding a constant \(b\) to a random variable does not change its variance, while multiplying by \(a\) scales the variance by \(a^2\).

• Variance of the Sum/Difference of Two Independent Random Variables: If \(X\) and \(Y\) are independent, \[ \text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y) \]

Named Distributions

• For a Binomial distribution, understand when it applies (BInS criteria) and how to calculate probabilities, expected values, and variances: \[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}\] \[\text{E}(X) = np, ~~~\sigma_X = \sqrt{np(1-p)}\]

• For a Poisson distribution, recognize when it applies and how to calculate probabilities, expected values, and variances: \[P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}\] \[\text{E}(X) = \lambda, ~~\sigma_X = \sqrt{\lambda}\]

Chapter 6: Continuous Random Variables and Probability Distributions

• Determine if a function is a legitimate density function and calculate the normalization constant if necessary.

  • Legitimate Density Functions: A function \(f(x)\) is a legitimate density function if it satisfies two conditions:
    1. \(f(x) \geq 0\) for all \(x\).
    2. The total area under the curve of \(f(x)\) over its entire range equals 1, i.e., \(\int_{-\infty}^{\infty} f(x) dx = 1\).
  • Normalization Constant: The constant required to ensure the total area under the probability density function (pdf) equals 1.

• Calculate probabilities for a continuous random variable using the density function: \[P(a < X < b) = \int_a^b f(x)dx\]

• Calculate and use the cumulative distribution function (CDF): \[F(x) = P(X \leq x) = \int_{-\infty}^x f(t)dt\]

• Percentiles and Median:

  • Percentile: Solve \(F(y) = p\) to find the \(y\)th percentile.
  • Median: \[\int_{-\infty}^{\tilde{\mu}} f(x) dx = 0.5\]
  • Mean (Expected Value): \[E(X) = \mu_X = \int_{-\infty}^{\infty} x \cdot f(x)dx\]

Named Distributions

(Note: The distributions have been written in short hand notation. You need to realize where the pdf/cdf is 0 and where the cdf is 1.)

• Normal Distribution:
  • Use the z-table for calculating probabilities and percentiles.
  • Normal probability plots help determine if data follow a normal distribution. Deviations suggest skewness or a non-normal distribution.
• For a Uniform Distribution, understand when it applies and how to calculate probabilities, percentiles, expected values, and variances:
  • Probability Density Function (pdf): \[f(x) = \frac{1}{b-a}, ~~~ \text{for} ~~ a \leq x < b\]
  • CDF: \[F(x) = \frac{x-a}{b-a}, ~~~ \text{for} ~~ a \leq x < b\]
  • Mean and Standard Deviation: \[E(X) = \frac{a+b}{2},~~~ \sigma = \sqrt{\frac{(b-a)^2}{12}}\]
• For a Exponential Distribution, understand when it applies and how to calculate probabilities, percentiles, expected values, and variances:
  • pdf: \[f(x) = \lambda e^{-\lambda x}, ~~~ \text{for} ~~ x\geq 0\]
  • CDF: \[F(x) = 1 - e^{-\lambda x}, ~~~ \text{for} ~~ x\geq 0\]
  • Mean and Standard Deviation: \[E(X) = \frac{1}{\lambda},~~~ \sigma = \frac{1}{\lambda}\]