26
Apr 23

## The magic of the distribution function

Let $X$ be a random variable. The function $F_{X}\left( x\right) =P\left(X\leq x\right) ,$ where $x$ runs over real numbers, is called a distribution function of $X.$ In statistics, many formulas are derived with the help of $F_{X}\left( x\right) .$ The motivation and properties are given here

Oftentimes, working with the distribution function is an intermediate step to obtain a density $f_{X}$ using the link

$F_{X}\left( x\right) =\int_{-\infty }^{x}f_{X}\left( t\right) dt.$

A series of exercises below show just how useful the distribution function is.

Exercise 1. Let $Y$ be a linear transformation of $X,$ that is, $Y=\sigma X+\mu ,$ where $\sigma >0$ and $\mu \in R.$ Find the link between $F_{X}$ and $F_{Y}.$ Find the link between $f_{X}$ and $f_{Y}.$
The solution is here.

The more general case of a nonlinear transformation can also be handled:

Exercise 2. Let $Y=g\left( X\right)$ where $g$ is a deterministic function. Suppose that $g$ is strictly  monotone, differentiable and $g^{-1}$ exists. Find the link between $F_{X}$ and $F_{Y}.$ Find the link between $f_{X}$ and $f_{Y}.$

Solution. The result differs depending on whether $g$ is increasing or decreasing. Let's assume the latter, so that $x_{1}\leq x_{2}$ is equivalent to $g\left( x_{1}\right) \geq g\left( x_{2}\right) .$ Also for simplicity suppose that $P\left( X=c\right) =0$ for any $c\in R.$ Then

$F_{Y}\left( y\right) =P\left( g\left( X\right) \leq y\right) =P\left( X\geq g^{-1}\left( y\right) \right) =1-P\left( X\leq g^{-1}\left( y\right) \right)=1-F_{X}\left( g^{-1}\left( y\right) \right) .$

Differentiation of this equation produces

$f_{Y}\left( y\right) =-f_{X}\left( g^{-1}\left( y\right) \right) \left( g^{-1}\left( y\right) \right) ^{\prime }=f_{X}\left( g^{-1}\left( y\right) \right) \left\vert \left( g^{-1}\left( y\right) \right) ^{\prime }\right\vert$

(the derivative of $g^{-1}$ is negative).

For an example when $g$ is not invertible see the post about the chi-squared distribution.

Exercise 3. Suppose $T=X/Y$ where $X$ and $Y$ are independent, have densities $f_{X},f_{Y}$ and $Y>0.$ What are the distribution function and density of $T?$

Solution. The joint density $f_{X,Y}$ equals $f_{X}f_{Y},$ so

$F_{T}\left( t\right) =P\left( T\leq t\right) =P\left( X\leq tY\right) =\underset{x\leq ty}{\int \int }f_{X}\left( x\right) f_{Y}\left( y\right) dxdy$

(converting a double integral to an iterated integral and remembering that $f_{Y}$ is zero on the left half-axis)

$=\int_{0}^{\infty }\left( \int_{-\infty }^{ty}f_{X}\left( x\right) dx\right) f_{Y}\left( y\right) dy=\int_{0}^{\infty }F_{X}\left( ty\right) f_{Y}\left( y\right) dy.$

Now by the Leibniz integral rule

(1) $f_{T}\left( t\right) =\int_{0}^{\infty }f_{X}\left( ty\right) yf_{Y}\left( y\right) dy.$

A different method is indicated in Activity 4.11, p.207 of J. Abdey, Guide ST2133.

Exercise 4. Let $X,Y$ be two independent random variables with densities $f_{X},f_{Y}$. Find $F_{X+Y}$ and $f_{X+Y}.$

See this post.

Exercise 5. Let $X,Y$ be two independent random variables. Find $F_{\max\left\{ X_{1},X_{2}\right\} }$ and $F_{\min \left\{ X_{1},X_{2}\right\} }.$

Solution. The inequality $\max \left\{ X_{1},X_{2}\right\} \leq x$ holds if and only if both $X_{1}\leq x$ and $X_{2}\leq x$ hold. This means that the event $\left\{ \max \left\{ X_{1},X_{2}\right\} \leq x\right\}$ coincides with the event $\left\{ X_{1}\leq x\right\} \cap \left\{ X_{2}\leq x\right\} .$ It follows by independence that

$F_{\max \left\{ X_{1},X_{2}\right\}}\left( x\right) =P\left( \max \left\{ X_{1},X_{2}\right\} \leq x\right) =P\left( \left\{ X_{1}\leq x\right\} \cap \left\{ X_{2}\leq x\right\}\right)$

$=PX_{1}\leq x)P\left( X_{2}\leq x\right) =F_{X_{1}}\left( x\right)F_{X_{2}}\left( x\right) .$

For $\min \left\{ X_{1},X_{2}\right\}$ we need one more trick - pass to the complementary event by writing $F_{\min \left\{ X_{1},X_{2}\right\} }\left( x\right) =P\left( \min \left\{ X_{1},X_{2}\right\} \leq x\right) =1-P\left(\min \left\{ X_{1},X_{2}\right\} >x\right) .$ Now we can use the fact that the event $\left\{ \min \left\{ X_{1},X_{2}\right\} >x\right\}$ coincides with the event $\left\{ X_{1}>x\right\} \cap \left\{ X_{2}>x\right\} .$ Hence, by independence

$F_{\min \left\{ X_{1},X_{2}\right\} }\left( x\right) =1-P\left( \left\{ X_{1}>x\right\} \cap \left\{ X_{2}>x\right\} \right) =1-P\left( X_{1}>x\right) P\left( X_{2}>x\right)$

$=1-\left[ 1-P\left( X_1\leq x\right) \right] \left[ 1-P\left( X_2\leq x\right) \right] =1-\left( 1-F_{X_1}\left( x\right) \right) \left( 1-F_{X_2}\left( x\right) \right) .$

27
Dec 22

## Final exam in Advanced Statistics ST2133, 2022

Unlike most UoL exams, here I tried to relate the theory to practical issues.

KBTU International School of Economics

Compiled by Kairat Mynbaev

The total for this exam is 41 points. You have two hours.

Everywhere provide detailed explanations. When answering please clearly indicate question numbers. You don’t need a calculator. As long as the formula you provide is correct, the numerical value does not matter.

Question 1. (12 points)

a) (2 points) At a casino, two players are playing on slot machines. Their payoffs $X,Y$ are standard normal and independent. Find the joint density of the payoffs.

b) (4 points) Two other players watch the first two players and start to argue what will be larger: the sum $U = X + Y$ or the difference $V = X - Y$. Find the joint density. Are variables $U,V$ independent? Find their marginal densities.

c) (2 points) Are $U,V$ normal? Why? What are their means and variances?

d) (2 points) Which probability is larger: $P(U > V)$ or $P\left( {U < V} \right)$?

e) (2 points) In this context interpret the conditional expectation $E\left( {U|V = v} \right)$. How much is it?

Reminder. The density of a normal variable $X \sim N\left( {\mu ,{\sigma ^2}} \right)$ is ${f_X}\left( x \right) = \frac{1}{{\sqrt {2\pi {\sigma ^2}} }}{e^{ - \frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}$.

Question 2. (9 points) The distribution of a call duration $X$ of one Kcell [largest mobile operator in KZ] customer is exponential: ${f_X}\left( x \right) = \lambda {e^{ - \lambda x}},\,\,x \ge 0,\,\,{f_X}\left( x \right) = 0,\,\,x < 0.$ The number $N$ of customers making calls simultaneously is distributed as Poisson: $P\left( {N = n} \right) = {e^{ - \mu }}\frac{{{\mu ^n}}}{{n!}},\,\,n = 0,1,2,...$ Thus the total call duration for all customers is ${S_N} = {X_1} + ... + {X_N}$ for $N \ge 1$. We put ${S_0} = 0$. Assume that customers make their decisions about calling independently.

a) (3 points) Find the general formula (when ${X_1},...,{X_n}$ are identically distributed and $X,N$ are independent but not necessarily exponential and Poisson, as above) for the moment generating function of $S_N$ explaining all steps.

b) (3 points) Find the moment generating functions of $X$, $N$ and ${S_N}$ for your particular distributions.

c) (3 points) Find the mean and variance of ${S_N}$. Based on the equations you obtained, can you suggest estimators of parameters $\lambda ,\mu$?

Remark. Direct observations on the exponential and Poisson distributions are not available. We have to infer their parameters by observing ${S_N}$. This explains the importance of the technique used in Question 2.

Question 3. (8 points)

a) (2 points) For a non-negative random variable $X$ prove the Markov inequality $P\left( {X > c} \right) \le \frac{1}{c}EX,\,\,\,c > 0.$

b) (2 points) Prove the Chebyshev inequality $P\left( {|X - EX| > c} \right) \le \frac{1}{c^2}Var\left( X \right)$ for an arbitrary random variable $X$.

c) (4 points) We say that the sequence of random variables $\left\{ X_n \right\}$ converges in probability to a random variable $X$ if $P\left( {|{X_n} - X| > \varepsilon } \right) \to 0$ as $n \to \infty$ for any $\varepsilon > 0$.  Suppose that $E{X_n} = \mu$ for all $n$ and that $Var\left(X_n \right) \to 0$ as $n \to \infty$. Prove that then $\left\{X_n\right\}$ converges in probability to $\mu$.

Remark. Question 3 leads to the simplest example of a law of large numbers: if $\left\{ X_n \right\}$ are i.i.d. with finite variance, then their sample mean converges to their population mean in probability.

Question 4. (8 points)

a) (4 points) Define a distribution function. Give its properties, with intuitive explanations.

b) (4 points) Is a sum of two distribution functions a distribution function? Is a product of two distribution functions a distribution function?

Remark. The answer for part a) is here and the one for part b) is based on it.

Question 5. (4 points) The Rakhat factory prepares prizes for kids for the upcoming New Year event. Each prize contains one type of chocolates and one type of candies. The chocolates and candies are chosen randomly from two production lines, the total number of items is always 10 and all selections are equally likely.

a) (2 points) What proportion of prepared prizes contains three or more chocolates?

b) (2 points) 100 prizes have been sent to an orphanage. What is the probability that 50 of those prizes contain no more than two chocolates?

24
Oct 22

## A problem to do once and never come back

There is a problem I gave on the midterm that does not require much imagination. Just know the definitions and do the technical work, so I was hoping we could put this behind us. Turned out we could not and thus you see this post.

Problem. Suppose the joint density of variables $X,Y$ is given by

$f_{X,Y}(x,y)=\left\{\begin{array}{c}k\left( e^{x}+e^{y}\right) \text{ for }0\le y\le x\le 1, \\ 0\text{ otherwise.}\end{array}\right.$

I. Find $k$.

II. Find marginal densities of $X,Y$. Are $X,Y$ independent?

III. Find conditional densities $f_{X|Y},\ f_{Y|X}$.

IV. Find $EX,\ EY$.

When solving a problem like this, the first thing to do is to give the theory. You may not be able to finish without errors the long calculations but your grade will be determined by the beginning theoretical remarks.

### I. Finding the normalizing constant

Any density should satisfy the completeness axiom: the area under the density curve (or in this case the volume under the density surface) must be equal to one: $\int \int f_{X,Y}(x,y)dxdy=1.$ The constant $k$ chosen to satisfy this condition is called a normalizing constant. The integration in general is over the whole plain $R^{2}$ and the first task is to express the above integral as an iterated integral. This is where the domain where the density is not zero should be taken into account. There is little you can do without geometry. One example of how to do this is here.

The shape of the area $A=\left\{ (x,y):0 is determined by a) the extreme values of $x,y$ and b) the relationship between them. The extreme values are 0 and 1 for both $x$ and $y$, meaning that $A$ is contained in the square $\left\{ (x,y):0 The inequality $y means that we cut out of this square the triangle below the line $y=x$ (it is really the lower triangle because if from a point on the line $y=x$ we move down vertically, $x$ will stay the same and $y$ will become smaller than $x$).

In the iterated integral:

a) the lower and upper limits of integration for the inner integral are the boundaries for the inner variable; they may depend on the outer variable but not on the inner variable.

b) the lower and upper limits of integration for the outer integral are the extreme values for the outer variable; they must be constant.

This is illustrated in Pane A of Figure 1.

Figure 1. Integration order

Always take the inner integral in parentheses to show that you are dealing with an iterated integral.

a) In the inner integral integrating over $x$ means moving along blue arrows from the boundary $x=y$ to the boundary $x=1.$ The boundaries may depend on $y$ but not on $x$ because the outer integral is over $y.$

b) In the outer integral put the extreme values for the outer variable. Thus,

$\underset{A}{\int \int }f_{X,Y}(x,y)dxdy=\int_{0}^{1}\left(\int_{y}^{1}f_{X,Y}(x,y)dx\right) dy.$

Check that if we first integrate over $y$ (vertically along red arrows, see Pane B in Figure 1) then the equation

$\underset{A}{\int \int }f_{X,Y}(x,y)dxdy=\int_{0}^{1}\left(\int_{0}^{x}f_{X,Y}(x,y)dy\right) dx$

results.

In fact, from the definition $A=\left\{ (x,y):0 one can see that the inner interval for $x$ is $\left[ y,1\right]$ and for $y$ it is $\left[ 0,x\right] .$

### II. Marginal densities

The condition for independence of $X,Y$ is

$f_{X,Y}\left(x,y\right)=f_{X}\left( x\right) f_{Y}\left(y\right)$

(this is a direct analog of the independence condition for events $P\left(A\cap B\right) =P\left(A\right) P\left(B\right)$). In words: the joint density decomposes into a product of individual densities.

### III. Conditional densities

In this case the easiest is to recall the definition of conditional probability $P\left( A|B\right) =\frac{P\left( A\cap B\right) }{P\left(B\right) }.$ The definition of conditional densities $f_{X|Y},\ f_{Y|X}$ is quite similar:

(2) $f_{X|Y}\left( x|y\right) =\frac{f_{X,Y}\left( x,y\right) }{f_{Y}\left(y\right) },\ f_{Y|X}\left( y|x\right) =\frac{f_{X,Y}\left( x,y\right) }{f_{X}\left( x\right) }$.

Of course, $f_{Y}\left( y\right) ,f_{X}\left( x\right)$ here can be replaced by their marginal equivalents.

### IV. Finding expected values of $X,Y$$X,Y$

The usual definition $EX=\int xf_{X}\left( x\right) dx$ takes an equivalent form using the marginal density:

$EX=\int x\left( \int f_{X,Y}\left( x,y\right) dy\right) dx=\int \int xf_{X,Y}\left( x,y\right) dydx.$

Which equation to use is a matter of convenience.

Another replacement in the usual definition gives the definition of conditional expectations:

$E\left( X|Y\right) =\int xf_{X|Y}\left( x|y\right) dx,$ $E\left( Y|X\right) =\int yf_{Y|X}\left( y|x\right) dx.$

Note that these are random variables: $E\left( X|Y=y\right)$ depends in $y$ and $E\left( Y|X=x\right)$ depends on $x.$

### Solution to the problem

Being a lazy guy, for the problem this post is about I provide answers found in Mathematica:

I. $k=0.581977$

II. $f_{X}\left( x\right) =-1+e^{x}\left( 1+x\right) ,$ for $x\in[ 0,1],$ $f_{Y}\left( y\right) =e-e^{y}y,$ for $y\in \left[ 0,1\right] .$

It is readily seen that the independence condition is not satisfied.

III. $f_{X|Y}\left( x|y\right) =\frac{k\left( e^{x}+e^{y}\right) }{e-e^{y}y}$ for $0

$f_{Y|X}\left(y|x\right) =\frac{k\left(e^x+e^y\right) }{-1+e^x\left( 1+x\right) }$ for $0

IV. $EX=0.709012,$ $EY=0.372965.$

24
Oct 22

## Marginal probabilities and densities

This is to help everybody, from those who study Basic Statistics up to Advanced Statistics ST2133.

### Discrete case

Suppose in a box we have coins and banknotes of only two denominations: $1 and$5 (see Figure 1).

Figure 1. Illustration of two variables

We pull one out randomly. The division of cash by type (coin or banknote) divides the sample space (shown as a square, lower left picture) with probabilities $p_{c}$ and $p_{b}$ (they sum to one). The division by denomination ($1 or$5) divides the same sample space differently, see the lower right picture, with the probabilities to pull out $1 and$5 equal to $p_{1}$ and $p_{5}$, resp. (they also sum to one). This is summarized in the tables

 Variable 1: Cash type Prob coin $p_{c}$$p_{c}$ banknote $p_{b}$$p_{b}$
 Variable 2: Denomination Prob $1 $p_{1}$$p_{1}$$5 $p_{5}$$p_{5}$

Now we can consider joint events and probabilities (see Figure 2, where the two divisions are combined).

Figure 2. Joint probabilities

For example, if we pull out a random $item$ it can be a $coin$ and \$1 and the corresponding probability is $P\left(item=coin,\ item\ value=\1\right) =p_{c1}.$ The two divisions of the sample space generate a new division into four parts. Then geometrically it is obvious that we have four identities:

Adding over denominations: $p_{c1}+p_{c5}=p_{c},$ $p_{b1}+p_{b5}=p_{b},$

Adding over cash types: $p_{c1}+p_{b1}=p_{1},$ $p_{c5}+p_{b5}=p_{5}.$

Formally, here we use additivity of probability for disjoint events

$P\left( A\cup B\right) =P\left( A\right) +P\left( B\right) .$

In words: we can recover own probabilities of variables 1,2 from joint probabilities.

### Generalization

Suppose we have two discrete random variables $X,Y$ taking values $x_{1},...,x_{n}$ and $y_{1},...,y_{m},$ resp., and their own probabilities are $P\left( X=x_{i}\right) =p_{i}^{X},$ $P\left(Y=y_{j}\right) =p_{j}^{Y}.$ Denote the joint probabilities $P\left(X=x_{i},Y=y_{j}\right) =p_{ij}.$ Then we have the identities

(1) $\sum_{j=1}^mp_{ij}=p_{i}^{X},$ $\sum_{i=1}^np_{ij}=p_{j}^{Y}$ ($n+m$ equations).

In words: to obtain the marginal probability of one variable (say, $Y$) sum over the values of the other variable (in this case, $X$).

The name marginal probabilities is used for $p_{i}^{X},p_{j}^{Y}$ because in the two-dimensional table they arise as a result of summing table entries along columns or rows and are displayed in the margins.

### Analogs for continuous variables with densities

Suppose we have two continuous random variables $X,Y$ and their own densities are $f_{X}$ and $f_{Y}.$ Denote the joint density $f_{X,Y}$. Then replacing in (1) sums by integrals and probabilities by densities we get

(2) $\int_R f_{X,Y}\left( x,y\right) dy=f_{X}\left( x\right) ,\ \int_R f_{X,Y}\left( x,y\right) dx=f_{Y}\left( y\right) .$

In words: to obtain one marginal density (say, $f_{Y}$) integrate out the other variable (in this case, $x$).

27
Jan 21

## AP Stats and Business Stats

Its content, organization and level justify its adoption as a textbook for introductory statistics for Econometrics in most American or European universities. The book's table of contents is somewhat standard, the innovation comes in a presentation that is crisp, concise, precise and directly relevant to the Econometrics course that will follow. I think instructors and students will appreciate the absence of unnecessary verbiage that permeates many existing textbooks.

Having read Professor Mynbaev's previous books and research articles I was not surprised with his clear writing and precision. However, I was surprised with an informal and almost conversational one-on-one style of writing which should please most students. The informality belies a careful presentation where great care has been taken to present the material in a pedagogical manner.

Carlos Martins-Filho
Professor of Economics
Boulder, USA

26
May 20

2
Mar 20

## Statistical calculator

In my book I explained how one can use Excel to do statistical simulations and replace statistical tables commonly used in statistics courses. Here I go one step further by providing a free statistical calculator that replaces the following tables from the book by Newbold et al.:

Table 1 Cumulative Distribution Function, F(z), of the Standard Normal Distribution Table

Table 2 Probability Function of the Binomial Distribution

Table 5 Individual Poisson Probabilities

Table 7a Upper Critical Values of Chi-Square Distribution with $\nu$ Degrees of Freedom

Table 8 Upper Critical Values of Student’s t Distribution with $\nu$ Degrees of Freedom

Tables 9a, 9b Upper Critical Values of the F Distribution

The calculator is just a Google sheet with statistical functions, see Picture 1:

Picture 1. Calculator using Google sheet

## How to use Calculator

1. Open an account at gmail.com, if you haven't already. Open Google Drive.
3. Find the sheet on my Google drive and copy it to your Google drive (File/Make a copy). An icon of my calculator will appear in your drive. That's not the file, it's just a link to my file. To the right of it there are three dots indicating options. One of them is "Make a copy", so use that one. The copy will be in your drive. After that you can delete the link to my file. You might want to rename "Copy of Calculator" as "Calculator".
5. When you click a cell, you can enter what you need either in the formula bar at the bottom or directly in the cell. You can also see the functions I embedded in the sheet.
6. In cell A1, for example, you can enter any legitimate formula with numbers, arithmetic signs, and Google sheet functions. Be sure to start it with =,+ or - and to press the checkmark on the right of the formula bar after you finish.
7. The cells below A1 replace the tables listed above. Beside each function there is a verbal description and further to the right - a graphical illustration (which is not in Picture 1).
8. On the tab named Regression you can calculate the slope and intercept. The sample size must be 10.
9. Keep in mind that tables for continuous distributions need two functions. For example, in case of the standard normal distribution one function allows you to go from probability (area of the left tail) to the cutting value on the horizontal axis. The other function goes from the cutting value on the horizontal axis to probability.
10. Feel free to add new sheets or functions as you may need. You will have to do this on a tablet or computer.
18
Nov 19

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 $\sigma =2,$ $\mu =3.$

16
Mar 19

## AP Statistics the Genghis Khan way 1

Recently I enjoyed reading Jack Weatherford's "Genghis Khan and the Making of the Modern World" (2004). I was reading the book with a specific question in mind: what were the main reasons of the success of the Mongols? Here you can see the list of their innovations, some of which were in fact adapted from the nations they subjugated. But what was the main driving force behind those innovations? The conclusion I came to is that Genghis Khan was a genial psychologist. He used what he knew about individual and social psychology to constantly improve the government of his empire.

I am no Genghis Khan but I try to base my teaching methods on my knowledge of student psychology.

### Problems and suggested solutions

Steven Krantz in his book (How to teach mathematics : Second edition, 1998, don't remember the page) says something like this: If you want your students to do something, arrange your classes so that they do it in the class.

Problem 1. Students mechanically write down what the teacher says and writes.

Solution. I don't allow my students to write while I am explaining the material. When I explain, their task is to listen and try to understand. I invite them to ask questions and prompt me to write more explanations and comments. After they all say "We understand", I clean the board and then they write down whatever they understood and remembered.

Problem 2. Students are not used to analyze what they read or write.

Solution. After students finish their writing, I ask them to exchange notebooks and check each other's writings. It's easier for them to do this while everything is fresh in their memory. I bought and distributed red pens. When they see that something is missing or wrong, they have to write in red. Errors or omissions must stand out. Thus, right there in the class students repeat the material twice.

Problem 3. Students don't study at home.

Solution. I let my students know in advance what the next quiz will be about. Even with this knowledge, most of them don't prepare at home. Before the quiz I give them about half an hour to repeat and discuss the material (this is at least the third repetition). We start the quiz when they say they are ready.

Problem 4. Students don't understand that active repetition (writing without looking at one's notes) is much more productive than passive repetition (just reading the notes).

Solution. Each time before discussion sessions I distribute scratch paper and urge students to write, not just read or talk. About half of them follow my recommendation. Their desire to keep their notebooks neat is not their last consideration. The solution to Problem 1 also hinges upon active repetition.

Problem 5. If students work and are evaluated individually, usually there is no or little interaction between them.

Solution. My class is divided in teams (currently I have teams of two to six people). I randomly select one person from each team to write the quiz. That person's grade is the team's grade. This forces better students to coach others and weaker students to seek help.

Problem 6. Some students don't want to work in teams. They are usually either good students, who don't want to suffer because of weak team members, or weak students, who don't want their low grades to harm other team members.

Solution. The good students usually argue that it's not fair if their grade becomes lower because of somebody else's fault. My answer to them is that the meaning of fairness depends on the definition. In my grading scheme, 30 points out of 100 is allocated for team work and the rest for individual achievements. Therefore I never allow good students to work individually. I want them to be my teaching assistants and help other students. While doing so, I tell them that I may reward good students with a bonus in the end of the semester. In some cases I allow weak students to write quizzes individually but only if the team so requests. The request of the weak student doesn't matter. The weak student still has to participate in team discussions.

Problem 7. There is no accumulation of theoretical knowledge (flat learning curve).

Solution. a) Most students come from high school with little experience in algebra. I raise the level gradually and emphasize understanding. Students never see multiple choice questions in my classes. They also know that right answers without explanations will be discarded.

b) Normally, during my explanations I fill out the board. The amount of the information the students have to remember is substantial and increases over time. If you know a better way to develop one's internal vision, let me know.

c) I don't believe in learning the theory by doing applied exercises. After explaining the theory I formulate it as a series of theoretical exercises. I give the theory in large, logically consistent blocks for students to see the system. Half of exam questions are theoretical (students have to provide proofs and derivations) and the other half - applied.

d) The right motivation can be of two types: theoretical or applied, and I never substitute one for another.

Problem 8. In low-level courses you need to conduct frequent evaluations to keep your students in working shape. Multiply that by the number of students, and you get a serious teaching overload.

Solution. Once at a teaching conference in Prague my colleague from New York boasted that he grades 160 papers per week. Evaluating one paper per team saves you from that hell.

### Outcome

In the beginning of the academic year I had 47 students. In the second semester 12 students dropped the course entirely or enrolled in Stats classes taught by other teachers. Based on current grades, I expect 15 more students to fail. Thus, after the first year I'll have about 20 students in my course (if they don't fail other courses). These students will master statistics at the level of my book.