Lecture Notes - Probability for Continuous Variables

PSYC 210: Introduction to Data Analysis - Lecture 5: Probability for Continuous Variables

Table of Contents

1. Continuous vs. Discrete Variables
2. Probability Density Function (PDF)
3. Properties of Normal Distribution
4. Z-scores and their Usage
5. Continuous Probability Practice Problems

1. Continuous vs. Discrete Variables

• Definition: Continuous variables can take on any value within a given range, in contrast to discrete variables which are limited to specific values (e.g., integers).
  • Example: A continuous variable might be a temperature reading (e.g., 1.80, 3.56).
• Probability Assessment:
  • Continuous probabilities are typically assessed in terms of ranges rather than specific values.
  • For example, instead of calculating $P(X = x<em>i$), we assess probabilities as $P(X \leq x</em>i)$ or $P(X \geq x_i)$.

2. Probability Density Function (PDF)

Definition

• PDF: Unlike discrete variables that are defined by probability mass functions (PMFs), continuous variables use probability density functions (PDFs).
• Visual Distinction:
  • PMFs resemble histograms (bars representing probabilities).
  • PDFs are represented as curves.

Equation

• The general form of a PDF is given by:
  $f(X | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{\frac{-(X-\mu)^2}{2\sigma^2}}$
• Note: While this equation represents Gaussian distributions, it is not crucial for students in this course to memorize it.

Purpose of PDF

• PDFs help to estimate the probability of the value of a continuous random variable falling within a certain range—a geometric shape representing areas (like a pie chart), where the area is equivalent to probability.
• Total Probability: $P(S) = 1$, indicating that the total area under the PDF should equal 1.

3. Properties of Normal Distribution

• The normal distribution, a commonly used PDF, possesses several key properties:
  • Symmetry: It is symmetrical around the mean.
  • Mesokurtic: The kurtosis (g2) is 0, indicating a standard peak.
  • Unimodal: It has one peak.
  • Mean, Median, Mode: All are equal ($\mu = Median = Mode$).
  • Inflection Points: Occur at $\mu \pm \sigma$, indicating where the curve bends.
  • 68-95-99.7 Rule:
  • Approximately 68% of data falls within one standard deviation: $\mu \pm \sigma$.
  • Approximately 95% falls within two standard deviations: $\mu \pm 2\sigma$.
• Notation: The normal distribution is denoted as $X \sim N(\mu, \sigma^2)$.

4. Z-scores and their Usage

• Definition: Z-scores are standardized scores indicating how many standard deviations a specific data point is from the mean.
• Transformation:
  • A raw observed score ($X<em>i$) can be transformed into a z-score using the equation: $z</em>i = \frac{x_i - \mu}{\sigma}$
  • Conversely, $x<em>i$ can be retrieved from $z</em>i$:
    $x<em>i = z</em>i(\sigma) + \mu$

Application Scenario

• For example, researchers from different regions might measure temperatures in various units (Celsius, Fahrenheit, Kelvin). Using z-scores allows for the normalization of these measurements into a common metric.

Z-distribution

• The z-distribution is the normal distribution of z-scores and is used for calculating probabilities associated with z-scores, represented in a z-table.
• The z-table provides the area under the curve for specific z-scores, which can be used to find probabilities for various scenarios.

5. Continuous Probability Practice Problems

Assume mean ($\mu$) = 35 and standard deviation ($\sigma$) = 5 for the following problems:

1. Probability of scoring 31 or greater.
2. Probability of scoring 41.25 or greater.
3. Probability of scoring 35 or less.
4. Probability of scoring 24.5 or less.
5. Probability of scoring ≤ 24.5 or ≥ 40.
6. Probability of scoring between 30 and 40.
7. Probability of scoring between 32 and 36.5.

Note:

• For practice problems, utilize the properties of the normal distribution and refer to the appropriate z-scores to compute the probabilities. Always visualize the distribution to aid in understanding and solving the problems.