Day 8

Math 216: Statistical Thinking

Bastola

Introduction to Random Variables (RVs)

Overview

  • Random Variables (RVs): Represent outcomes of experiments, similar to variables in algebra.
  • Total Probability: Must sum to 1 across all sample points.
  • Sample Spaces:
    • Finite: Straightforward probability assignment.
    • Infinite: Require probability models for practical application.

Discrete Random Variables

Definition

Discrete Random Variables: Assume a countable number of values.

Examples of Discrete RVs

  1. Seizures per Week: Number of seizures a patient has (\(x=0, 1, 2, \ldots\)).
  2. Voters Favoring Impeachment: Out of a sample of 500 (\(x=0, 1, 2, \ldots, 500\)).
  3. Tennis Player Shoe Sizes: Possible shoe sizes (\(x=5, 5.5, 6, 6.5, \ldots\)).
  4. Change Received: Possible amounts of change given (\(x=1¢, 2¢, 3¢, \ldots, \$1, \$1.01, \$1.02, \ldots\)).
  5. Waiting Customers: Number of customers waiting in a restaurant (\(x=0, 1, 2, \ldots\)).

Continuous Random Variables

Definition

Continuous Random Variables: Assume values from any point within an interval.

Examples of Continuous RVs

  1. Time Between Hospital Arrivals: Time in seconds (\(0 \leq x < \infty\)).
  2. Time to Complete an Exam: Time in minutes for a one-hour exam (\(0 \leq x \leq 60\)).
  3. Beverage Amount in a Can: Ounces in a 12-ounce can (\(0 \leq x \leq 12\)).
  4. Oil-Drilling Depth: Depth in feet until oil is struck (\(0 \leq x \leq c\), where \(c\) is the max depth).
  5. Weight of Food Item: Weight in pounds of a supermarket item (\(0 \leq x \leq 500\)).

Probability Distributions for Discrete RVs

Definition

Probability Distribution: Specifies all possible values and the probabilities of a discrete random variable.

Example: Roll of a Die

  • Possible Values: \(x = 1, 2, 3, 4, 5, 6\)
  • Probability of Each Value: \(P(x = k) = \frac{1}{6}\) for \(k = 1, 2, 3, 4, 5, 6\)

This distribution fully describes the likelihood of each outcome when a fair die is rolled.

Key Requirements for Discrete Probability Distributions

Requirements

  • Probability Range: \(0 \leq p(x) \leq 1\) for each outcome \(x\).
  • Total Probability: Sum of all probabilities must be 1: \[ \sum_{\text{all } x} p(x) = 1 \]

Expected Value: Mean of Discrete Random Variables

Definition

The expected value (\(E(X)\)), or mean, is the weighted average of all possible values.

Formula

\[ \mu = E(X) = \sum_{\text{all } x_i} x_i p(x_i) = x_1 p(x_1) + x_2 p(x_2) + \cdots \]

Motivation

  • Represents the “center” or typical value you expect if the random experiment is repeated many times.

Variance and Standard Deviation

Definition

Variance measures the spread of values around the mean.

\[ \sigma^2 = \operatorname{Var}(x) = E\left[(x-\mu)^2\right] = \sum_{\text{all } x_i}\left(x_i-\mu\right)^2 p(x_i) = \left[\sum_{\text{all } x_i} x_i^2 p(x_i)\right] - \mu^2 \]

Standard Deviation is the square root of variance, indicating typical deviation from the mean.

\[ \sigma = \sqrt{\operatorname{Var}(x)} \]