This assignment is designed to review the materials you learn in the lab. Be sure to comment your code to clarify what you are doing. Not only does it help with grading, but it will also help you when you revisit it in the future. Please post any questions on Piazza.

Using the Normal Distribution

Let \(X \sim \mathcal{N}(-1,9)\) and \(Y \sim \mathcal{N}(10,4)\). Let \(z_i \in \mathbb{R}\) be the result of draws from \(X\) or \(Y\) with equal probability. That is, there is a 0.5 chance that \(z_i\) comes from \(X\) and 0.5 that it comes from \(Y\). We observe the value of \(z_i\) but are not told which of \(X\) or \(Y\) it came from.

1) Checking Intuition
Suppose \(z_1 = 0\) and \(z_2 = 15\).

  1. Is it more probable that \(z_1\) came from \(X\) or \(Y\)? Why?
  2. Is it more probable that \(z_2\) came from \(X\) or \(Y\)? Why?

2) Math

  1. Find \(\mathbb{P}(z_i \leq 0)\) if we assume \(z_i\) came from \(X\).
  2. Find \(\mathbb{P}(z_i \leq 0)\) if we assume \(z_i\) came from \(Y\).

Hint: both answers should be expressed as \(\Phi(\cdot)\).

3) Coding in R
Using base R or ggplot2, plot the probability density function and area under the curve for \(z_i\) under the assumptions from 2a). Use pnorm() to report your result from 2a) numerically.

Summarizing Distributions

4) Checking Intuition
Think about a random variable \(X\) that you’re interested in for your research. Intuitively, how would we characterize the central tendency and spread of \(X\)?

5) Math
Let \(X\) and \(Y\) be random variables and \(a, b \in \mathbb{R}\). Using the definitions and properties of expectation and variance,
a) Show that \(\mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2\)
b) Show that \(\mathbb{V}(X + a) = \mathbb{V}(X)\)
c) Show that \(\mathbb{V}(a \cdot X) = a^2 \cdot \mathbb{V}(X)\)

6) Jordan’s Conjecture
For two random variables \(X\) and \(Y\), we say \(X\) and \(Y\) are independent, or \(X \perp Y\) if \(\mathbb{E}[X \cdot Y] = \mathbb{E}[X] \cdot \mathbb{E}[Y]\). Apparently, two years ago in this class on Zoom, I asked something pretty smart (I have absolutely no memory of doing this): is the reverse true? In other words, we know \(X \perp Y \implies \mathbb{E}[X \cdot Y] = \mathbb{E}[X] \cdot \mathbb{E}[Y]\), but is \(\mathbb{E}[X \cdot Y] = \mathbb{E}[X] \cdot \mathbb{E}[Y] \implies X \perp Y\) also true? If it is true, prove it. If not, provide a counterexample.