Gov 2002: Problem Set 5

Submission instructions | PDF | Rmd |

Problem Set Instructions

This problem set is due on October 25, 11:59 pm Eastern time. Please upload a PDF of your solutions to Gradescope. We will accept hand-written solutions but we strongly advise you to typeset your answers in Rmarkdown. Please list the names of other students you worked with on this problem set.

Question 1 (25 points)

Suppose we want to model the relationship between legislation and politician quality. There are two types of politician quality: high and low. When a high quality politician propose a bill, it has a probability $p_1$ to pass; conversely, when a low quality politician propose a bill, it has a probability $p_2$ to pass, where $p_1>p_2$. Unfortunately, we cannot directly observe politicians’ quality, but instead rely on our prior that a politician is a high type with probability $h$ and low type with probability $1-h$, where $h\in (0,1)$. Let $X$ be the number of passed bills after a randomly picked politician has made $n$ proposals.

1. Find the marginal distribution of $X$.

2. Find the mean and variance of $X$.

Question 2 (25 points)

We know from the definition of the variance that $\mathbb{E}\left[{(Y-\mathbb{E}[Y])}^2\right]=\mathbb{E}[Y^2]-{(\mathbb{E}[Y])}^2$. Prove that this equality still holds when we condition on $X$, i.e., $\mathbb{E}[{(Y-\mathbb{E}[Y\mid X])}^2\mid X]=\mathbb{E}[Y^2\mid X]-{(\mathbb{E}[Y\mid X])}^2$

Question 3 (30 points)

Let $X_1\dots X_n$ be i.i.d. r.v.s with mean $\mu$ and variance ${\sigma }^2$, and $n\ge 2$. A bootstrap sample of $X_1\dots X_n$ is a sample of $n$ r.v.s $X_1^{\ast }\dots X_n^{\ast }$ formed from the $X_j$ by sampling with replacement with equal probabilities. Let ${\overline{X}}^{\ast }$ denote the sample mean of the bootstrap sample:

$${\overline{X}}^{\ast }=\frac{1}{n}(X_1^{\ast }+\dots X_n^{\ast })$$

1. Find $\mathbb{E}[X_j^{\ast }]$ and $\mathbb{V}[X_j^{\ast }]$ for each $j$. (Hint: What is the distribution of $X_j^{\ast }$?)

2. Find $\mathbb{E}[{\overline{X}}^{\ast }\mid X_1,\dots ,X_n]$ and $\mathbb{V}[{\overline{X}}^{\ast }\mid X_1,\dots ,X_n]$ (Hint: Conditional on $X_1\dots X_n$, the $X_j^{\ast }$ are independent, with a PMF that puts probability $1/n$ at each of the points $X_1\dots X_n$.)

3. Find $\mathbb{E}\left[{\overline{X}}^{\ast }\right]$ and $\mathbb{V}\left[{\overline{X}}^{\ast }\right]$ (Hint: Recall that the sample variance $\frac{1}{n-1}\sum _{j=1}^n(X_j-\overline{X}{)}^2$ is an unbiased estimator of the population variance ${\sigma }^2$)

Question 4 (20 points)

Jon commutes on the Boston subway from Park Street Station to Harvard Square. He records in minutes every day how long he waits for the train to arrive. He assumes a statistical model that says his waiting times $Y_1\dots Y_n$ are i.i.d. from Unif(0, $\theta$).

1. Find an unbiased plug-in estimator ${\hat{\theta }}_{PI}$

2. Find the variance and mean square error of ${\hat{\theta }}_{PI}$