# Expected Value and Variance

STAT 20: Introduction to Probability and Statistics

## Agenda

• Concept Questions
• Handout: PS 4.2
• Break
• Lab 4

# Concept Questions

01:00

$X$ is a random variable with the distribution shown below:

$X = \begin{cases} 3, \; \text{ with prob } 1/3\\ 4, \; \text{ with prob } 1/4\\ 5, \; \text{ with prob } 5/12 \end{cases}$

Consider the box with tickets: $\fbox{3}\, \fbox{3}\, \fbox{3} \,\fbox{4} \,\fbox{4} \,\fbox{4} \,\fbox{4} \,\fbox{5} \,\fbox{5}\, \fbox{5} \,\fbox{5} \,\fbox{5}$ (Does this box represent $X$?)

The expected value of $X$ is ____ the average of the box. Fill in the blank with one of the choices listed below.

01:00

A die will be rolled $n$ times and the object is to guess the total number of spots in $n$ rolls, and you choose $n$ to be either 50 or 100. There is a one-dollar penalty for each spot that the guess is off. For instance, if you guess 200, and the total is 215, then you lose 15 dollars. Which do you prefer? 50 throws, or 100?

Which do you prefer? $n = 50$ rolls, or $n = 100$ rolls?

02:00

One hundred draws will be made with replacement from a box with tickets $\fbox{0}\, \fbox{2}\, \fbox{3} \,\fbox{4} \,\fbox{6}$. Which of the following statements are true?

• The expected value of the sum of the one hundred draws is 300, give or take 20 or so.
• The expected value of the sum of the one hundred is 300.
• The sum of the one hundred draws is 300, give or take 20 or so.
• The sum of the one hundred draws is 300.
01:00

We have two random variables: $X \sim$ Binomial($10, 0.2$) and $Y$ is the random variable that is the value of one ticket drawn at random from a box with tickets $\fbox{0}\, \fbox{2}\, \fbox{3} \,\fbox{4} \,\fbox{6}$.

We take the sum of 100 iid random variables for each of $X$ and $Y$, called $S_X$ and $S_Y$. The empirical distributions of $S_X$ and $S_Y$ are plotted below.

Which distribution belongs to which random variable?

# PS 4.2

20:00

# Break

05:00