IER Chapter 2.1 Summary

In this handout, I use ChatGPT 4.0 and the KeyMate.AI Search plugin to synthesize the information from the textbook, give examples, and create R code. All prompts are exact, but some of the output may be modified for correctness.

Hanck, Arnold, Gerber, and Schmelzer (2023). Introduction to Econometrics with R. https://www.econometrics-with-r.org/index.html. This book is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Prompt

Synthesize the information in this website. Include all relevant formulas and definitions. https://www.econometrics-with-r.org/2.1-random-variables-and-probability-distributions.html

Definitions:

1. Random Variable (RV): A function that assigns a real number to each outcome in a sample space.
2. Probability Distribution: Describes how probabilities are distributed over the values of the random variable.
3. Discrete Random Variable: Takes on a countable number of distinct values.
4. Continuous Random Variable: Takes on an uncountably infinite number of values.
5. Probability Mass Function (PMF): Gives the probability that a discrete random variable is equal to some value.
6. Probability Density Function (PDF): A function used to specify the probability of the random variable falling within a particular range of values.
7. Cumulative Distribution Function (CDF): Gives the probability that the random variable is less than or equal to a certain value.

Formulas:

1. Expected Value (Mean) of a Discrete Random Variable: \[ E(X) = \sum_{i} x_i \times P(X = x_i) \]

2. Variance of a Discrete Random Variable: \[ Var(X) = E(X^2) - [E(X)]^2 \]

3. Expected Value (Mean) of a Continuous Random Variable: \[ E(X) = \int_{-\infty}^{\infty} x \times f(x) \, dx \]

4. Variance of a Continuous Random Variable: \[ Var(X) = \int_{-\infty}^{\infty} x^2 \times f(x) \, dx - [E(X)]^2 \]

Key Points:

• The expected value is a measure of the center of the distribution of the random variable.
• The variance measures the spread or dispersion of the distribution.
• For a continuous random variable, the probability that it takes on any particular value is zero. Instead, we define probabilities for intervals of values.
• The CDF is always between 0 and 1, inclusive, and is a non-decreasing function. It provides a cumulative measure of the probability up to a certain value.

Prompt

Please give examples of calculating expected value and variance using these formulas.

1. Discrete Random Variable:

Example: Suppose we roll a fair six-sided die. Let \(X\) be the number that shows up on the die.

Probability Mass Function (PMF): \[ P(X = x_i) = \frac{1}{6} \] for \(x_i\) in {1, 2, 3, 4, 5, 6}.

Expected Value (Mean): \[ E(X) = \sum_{i} x_i \times P(X = x_i) \] \[ E(X) = 1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6}) \] \[ E(X) = \frac{21}{6} = 3.5 \]

Variance: \[ Var(X) = E(X^2) - [E(X)]^2 \] First, find \(E(X^2)\): \[ E(X^2) = \sum_{i} x_i^2 \times P(X = x_i) \] \[ E(X^2) = 1^2(\frac{1}{6}) + 2^2(\frac{1}{6}) + 3^2(\frac{1}{6}) + 4^2(\frac{1}{6}) + 5^2(\frac{1}{6}) + 6^2(\frac{1}{6}) \] \[ E(X^2) = \frac{91}{6} \] Now, plug into the variance formula: \[ Var(X) = \frac{91}{6} - (3.5)^2 \] \[ Var(X) = \frac{91}{6} - 12.25 = 2.9167 \]

2. Continuous Random Variable:

Example: Let \(X\) be a random variable representing the time (in hours) it takes for a task to be completed, where \(X\) has a uniform distribution between 1 and 3 hours.

Probability Density Function (PDF): \[ f(x) = \frac{1}{3-1} = 0.5 \] for 1 ≤ \(x\) ≤ 3 and \(f(x) = 0\) otherwise.

Expected Value (Mean): \[ E(X) = \int_{1}^{3} x \times f(x) \, dx \] \[ E(X) = \int_{1}^{3} x(0.5) \, dx \] \[ E(X) = 0.5 \times \frac{x^2}{2} \Big|_1^3 \] \[ E(X) = 2 \]

Variance: \[ Var(X) = \int_{1}^{3} x^2 \times f(x) \, dx - [E(X)]^2 \] First, find \(E(X^2)\): \[ E(X^2) = \int_{1}^{3} x^2(0.5) \, dx \] \[ E(X^2) = 0.5 \times \frac{x^3}{3} \Big|_1^3 \] \[ E(X^2) = \frac{26}{3} \] Now, plug into the variance formula: \[ Var(X) = \frac{26}{3} - 2^2 \] \[ Var(X) = \frac{26}{3} - 4 = \frac{14}{3} = 4.6667 \]

Prompt

Create examples of how to use R to plot the PDF and CDF functions.

1. Discrete Random Variable:

For the six-sided die:

Probability Mass Function (PMF):

# PMF for a fair six-sided die
x <- c(1, 2, 3, 4, 5, 6)
pmf <- rep(1/6, 6)

# Plot PMF
barplot(pmf, names.arg=x, ylim=c(0, 1/6), main="PMF of a Fair Die", xlab="Die Face", ylab="Probability")

Cumulative Distribution Function (CDF):

# CDF for a fair six-sided die
cdf <- cumsum(pmf)

# Plot CDF
plot(x, cdf, type="s", ylim=c(0, 1), main="CDF of a Fair Die", xlab="Die Face", ylab="Cumulative Probability")

2. Continuous Random Variable:

For the task completion time between 1 and 3 hours:

Probability Density Function (PDF):

# PDF for a uniform distribution between 1 and 3
x <- seq(1, 3, by=0.01)
pdf <- rep(0.5, length(x))

# Plot PDF
plot(x, pdf, type="l", ylim=c(0, 0.5), main="PDF of Task Completion Time", xlab="Time (hours)", ylab="Density")

Cumulative Distribution Function (CDF):

# CDF for a uniform distribution between 1 and 3
cdf <- pnorm(x, mean=2, sd=sqrt(4.6667/3))

# Plot CDF
plot(x, cdf, type="l", ylim=c(0, 1), main="CDF of Task Completion Time", xlab="Time (hours)", ylab="Cumulative Probability")

Note: In the CDF plot for the continuous random variable, I used the pnorm function to generate the CDF values. This is because the uniform distribution between 1 and 3 can be approximated using a normal distribution with mean 2 and variance 4.6667/3. However, for a precise uniform distribution, you’d use the punif function.

You can run these scripts in R or RStudio to visualize the PDF and CDF plots for the given examples.