17
Mar 19

## AP Statistics the Genghis Khan way 2

Last semester I tried to explain theory through numerical examples. The results were terrible. Even the best students didn't stand up to my expectations. The midterm grades were so low that I did something I had never done before: I allowed my students to write an analysis of the midterm at home. Those who were able to verbally articulate the answers to me received a bonus that allowed them to pass the semester.

This semester I made a U-turn. I announced that in the first half of the semester we will concentrate on theory and we followed this methodology. Out of 35 students, 20 significantly improved their performance and 15 remained where they were.

### Midterm exam, version 1

#### 1. General density definition (6 points)

a. Define the density $p_X$ of a random variable $X.$ Draw the density of heights of adults, making simplifying assumptions if necessary. Don't forget to label the axes.

b. According to your plot, how much is the integral $\int_{-\infty}^0p_X(t)dt?$ Explain.

c. Why the density cannot be negative?

d. Why the total area under the density curve should be 1?

e. Where are basketball players on your graph? Write down the corresponding expression for probability.

f. Where are dwarfs on your graph? Write down the corresponding expression for probability.

This question is about the interval formula. In each case students have to write the equation for the probability and the corresponding integral of the density. At this level, I don't talk about the distribution function and introduce the density by the interval formula.

#### 2. Properties of means (8 points)

a. Define a discrete random variable and its mean.

b. Define linear operations with random variables.

c. Prove linearity of means.

d. Prove additivity and homogeneity of means.

e. How much is the mean of a constant?

f. Using induction, derive the linearity of means for the case of $n$ variables from the case of two variables (3 points).

#### 3. Covariance properties (6 points)

a. Derive linearity of covariance in the first argument when the second is fixed.

b. How much is covariance if one of its arguments is a constant?

c. What is the link between variance and covariance? If you know one of these functions, can you find the other (there should be two answers)? (4 points)

#### 4. Standard normal variable (6 points)

a. Define the density $p_z(t)$ of a standard normal.

b. Why is the function $p_z(t)$ even? Illustrate this fact on the plot.

c. Why is the function $f(t)=tp_z(t)$ odd? Illustrate this fact on the plot.

d. Justify the equation $Ez=0.$

e. Why is $V(z)=1?$

f. Let $t>0.$ Show on the same plot areas corresponding to the probabilities $A_1=P(0 $A_2=P(z>t),$ $A_3=P(z<-t),$ $A_4=P(-t Write down the relationships between $A_1,...,A_4.$

#### 5. General normal variable (3 points)

a. Define a general normal variable $X.$

b. Use this definition to find the mean and variance of $X.$

c. Using part b, on the same plot graph the density of the standard normal and of a general normal with parameters $\sigma =2,$ $\mu =3.$

### Midterm exam, version 2

#### 1. General density definition (6 points)

a. Define the density $p_X$ of a random variable $X.$ Draw the density of work experience of adults, making simplifying assumptions if necessary. Don't forget to label the axes.

b. According to your plot, how much is the integral $\int_{-\infty}^0p_X(t)dt?$ Explain.

c. Why the density cannot be negative?

d. Why the total area under the density curve should be 1?

e. Where are retired people on your graph? Write down the corresponding expression for probability.

f. Where are young people (up to 25 years old) on your graph? Write down the corresponding expression for probability.

#### 2. Variance properties (8 points)

a. Define variance of a random variable. Why is it non-negative?

b. Define the formula for variance of a linear combination of two variables.

c. How much is variance of a constant?

d. What is the formula for variance of a sum? What do we call homogeneity of variance?

e. What is larger: $V(X+Y)$ or $V(X-Y)$? (2 points)

f. One investor has 100 shares of Apple, another - 200 shares. Which investor's portfolio has larger variability? (2 points)

#### 3. Poisson distribution (6 points)

a. Write down the Taylor expansion and explain the idea. How are the Taylor coefficients found?

b. Use the Taylor series for the exponential function to define the Poisson distribution.

c. Find the mean of the Poisson distribution. What is the interpretation of the parameter $\lambda$ in practice?

#### 4. Standard normal variable (6 points)

a. Define the density $p_z(t)$ of a standard normal.

b. Why is the function $p_z(t)$ even? Illustrate this fact on the plot.

c. Why is the function $f(t)=tp_z(t)$ odd? Illustrate this fact on the plot.

d. Justify the equation $Ez=0.$

e. Why is $V(z)=1?$

f. Let $t>0.$ Show on the same plot areas corresponding to the probabilities $A_1=P(0 $A_2=P(z>t),$ $A_{3}=P(z<-t),$ $A_4=P(-t Write down the relationships between $A_{1},...,A_{4}.$

#### 5. General normal variable (3 points)

a. Define a general normal variable $X.$

b. Use this definition to find the mean and variance of $X.$

c. Using part b, on the same plot graph the density of the standard normal and of a general normal with parameters $\sigma =2,$ $\mu =3.$