---
title: "Extra-Credit Quiz #3"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
---
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<!-- The code chunk below contains some settings that will  -->
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```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  fig.align="center",
  fig.pos="t",
  fig.width=12,
  strip.white = TRUE
)
```
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## Problem 1. **(3 points)** 
Let $X$ denote the number of $1's$ in 100 throws of a fair die. Find $\mathbb{E}[X^2].$


## Problem 2. **(2 points)** 
If $Var[X]=0$, then $\mathbb{P}[X = \mathbb{E}[X]] = 1.$ *True or false? You do not have to justify your response.*


## Problem 3. **(2 points)** 
Let $X$ denote the outcome of a roll of a fair, regular dodecahedron (a polyhedron with $12$ faces) with numbers $1,2, \cdots, 12$ written on its sides. Find $\mathbb{E}[X]$. 


## Problem 4. **(8 points)** 
Your sample consists of $50$ sixth-graders from Kealing Middle School (KMS) and $60$ sixth-graders from Murchison Middle school (MMS). The measures of center and spread of the students' heights are:
\begin{itemize}
  \item[$KMS:$] the mean of $60$ inches with the standard deviation of $1$ inch, and
  \item[$MMS:$] the mean of $60$ inches with the standard deviation of $2$ inches.
\end{itemize} 
What are the measures of center and spread for the pooled sample of $110$ sixth-graders?