Expected Value and Variance

STAT 20: Introduction to Probability and Statistics

Agenda

• Concept review
• Concept questions
• A new geometry: geom_col()
• PS 10
• Break
• Lab 4 slides
• Lab 4

Concept Review

Let \(X\) be a random variable such that \[ X = \begin{cases} -1, & \text{ with probability } 1/3\\ 0, & \text{ with probability } 1/6\\ 1, & \text{ with probability } 4/15 \\ 2, & \text{ with probability } 7/30 \\ \end{cases} \]

1. Draw the graph of the cdf of \(X\)

08:00

2. Compute the expected value and variance of \(X\)

write down pmf with denominator 30, and draw cdf on the board. P(X = -1)= 5/30, P(X = 0)= 10/30, P(X = 1)= 8/30, P(X = 2)= 7/30 E(X) = (-1)(5/30) + 0(10/30) + 1(8/30) + 2(7/30) (-10 + 0 + 8 + 14)/30 = 12/30 = 2/5 Var(X) = E(X^2) - 4/25 = (10 + 0 + 8 +28)/30 -4/25 = 23/15- 4/25 ~~ 1.37333 Graph the pmf and mark the expectation The numbers are not pretty but the point is to show them the computation and graph The rest of the ideas can be recapped as you go over the concept questions You may want to review linearity of expectation here, or save it for when going over the PS

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: \(\fbox{3}\, \fbox{3}\, \fbox{3} \,\fbox{4} \,\fbox{4} \,\fbox{4} \,\fbox{4} \,\fbox{5} \,\fbox{5}\, \fbox{5} \,\fbox{5} \,\fbox{5}\)

Suppose we draw once from this box and let \(Y\) be the value of the ticket drawn. Which random variable has a higher expectation?

The expected value of \(X\) is ____ the expected value of \(Y\).

Note that the pmf implied by the box is not the same as the given pmf. In the box, 4 has a higher probability, so the average is higher. They do not need to actually do the computation

Evgeni bakes a cake and invites two friends to join him in eating it. The cake will be evenly split between whoever shows up and Evgeni. His friends know that Evgeni is an erratic baker at best, and independently each toss a coin to decide whether they will go or not. Let \(X\) be the fraction of the cake that Evgeni gets to eat. For example if 1 friend shows up, Evgeni gets half the cake, if 2 show up, he gets one-third of the cake. What is \(E(X)\)?

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?

* From the text Statistics by Freedman, Pisani, and Purves

Prefer 50 because the square root law tells us the the spread of the distribution will be smaller for fewer rolls. If we were computing the average number of spots, then we would pick larger \(n\) since \(\sqrt{n}\) goes in the denominator of SE.

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.

* From the text Statistics by Freedman, Pisani, and Purves

The EV is a constant, and note that the box average is (0+2+3+4+6)/5 = 3, so the EV of a single draw is 3, and the EV of the sum of 100 draws is \(100\times 4 = 300\). The give or take number given here is the box SD \(\times 10 = 2\times 10 = 20\). You can let \(X\) be a random variable that is the value of one ticket drawn at random from this box, and then we want \(S_{100}\). This can be boardwork.

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 \(SX_{100}\) and \(SY_{100}\). The empirical distributions of \(SX_{100}\) and \(SY_{100}\) are plotted below.

Which distribution belongs to which random variable?

They need to remember that the expected value of S_x will be 200, and for S_y 300.

PS 10

20:00

Have them focus on working on problems 1, 2, 3 and attempt the last 2, and review the last 2 on the board.

Break

05:00

Visualizing probability distributions

Recall geom_bar(). What does it do? What aesthetics do we need to define?

ggplot(penguins, aes(x = species)) + 
geom_bar(fill = "steelblue4") + theme_minimal() 

What about if we don’t want to count instances but want to plot probabilities?

Let \(X\) be the number of heads when we toss a fair coin three times. We know that:

\[ X = \begin{cases} 0, & \text{ with probability } 1/8\\ 1, & \text{ with probability } 3/8\\ 2, & \text{ with probability } 3/8 \\ 3, & \text{ with probability } 1/8 \\ \end{cases} \] To plot this, we use a special kind of bar chart, plotted using geom_col(), in which the bar heights represent values supplied by the data. We have to specify a y aesthetic, a column from the data. The values from this column will be the heights of the bars.

x <- c(0, 1, 2, 3)
fx <- c(1/8, 3/8, 3/8, 1/8)
coin3_df <- data.frame(x, fx)
coin3_df %>% ggplot(aes(x = factor(x), y = fx)) +
geom_col(fill = "goldenrod2") + ???

Lab 4 (Slides)