Let be a random variable. The function where runs over real numbers, is called a distribution function of In statistics, many formulas are derived with the help of The motivation and properties are given here
Oftentimes, working with the distribution function is an intermediate step to obtain a density using the link
A series of exercises below show just how useful the distribution function is.
Exercise 1. Let be a linear transformation of that is, where and Find the link between and Find the link between and
The solution is here.
The more general case of a nonlinear transformation can also be handled:
Exercise 2. Let where is a deterministic function. Suppose that is strictly monotone, differentiable and exists. Find the link between and Find the link between and
Solution. The result differs depending on whether is increasing or decreasing. Let's assume the latter, so that is equivalent to Also for simplicity suppose that for any Then
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 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 or the difference . Find the joint density. Are variables independent? Find their marginal densities.
c) (2 points) Are normal? Why? What are their means and variances?
d) (2 points) Which probability is larger: or ?
e) (2 points) In this context interpret the conditional expectation . How much is it?
Reminder. The density of a normal variable is .
Question 2. (9 points) The distribution of a call duration of one Kcell [largest mobile operator in KZ] customer is exponential: The number of customers making calls simultaneously is distributed as Poisson: Thus the total call duration for all customers is for . We put . Assume that customers make their decisions about calling independently.
a) (3 points) Find the general formula (when are identically distributed and are independent but not necessarily exponential and Poisson, as above) for the moment generating function of explaining all steps.
b) (3 points) Find the moment generating functions of , and for your particular distributions.
c) (3 points) Find the mean and variance of . Based on the equations you obtained, can you suggest estimators of parameters ?
Remark. Direct observations on the exponential and Poisson distributions are not available. We have to infer their parameters by observing . This explains the importance of the technique used in Question 2.
Question 3. (8 points)
a) (2 points) For a non-negative random variable prove the Markov inequality
b) (2 points) Prove the Chebyshev inequality for an arbitrary random variable .
c) (4 points) We say that the sequence of random variables converges in probability to a random variable if as for any . Suppose that for all and that as . Prove that then converges in probability to .
Remark. Question 3 leads to the simplest example of a law of large numbers: if 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?
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 is given by
I. Find .
II. Find marginal densities of . Are independent?
III. Find conditional densities .
IV. Find .
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: The constant chosen to satisfy this condition is called a normalizing constant. The integration in general is over the whole plain 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 is determined by a) the extreme values of and b) the relationship between them. The extreme values are 0 and 1 for both and , meaning that is contained in the square The inequality means that we cut out of this square the triangle below the line (it is really the lower triangle because if from a point on the line we move down vertically, will stay the same and will become smaller than ).
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 means moving along blue arrows from the boundary to the boundary The boundaries may depend on but not on because the outer integral is over
b) In the outer integral put the extreme values for the outer variable. Thus,
Check that if we first integrate over (vertically along red arrows, see Pane B in Figure 1) then the equation
In fact, from the definition one can see that the inner interval for is and for it is
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 and (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 and , resp. (they also sum to one). This is summarized in the tables
Variable 1: Cash type
Variable 2: Denomination
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 it can be a and $1 and the corresponding probability is 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:
Adding over cash types:
Formally, here we use additivity of probability for disjoint events
In words: we can recover own probabilities of variables 1,2 from joint probabilities.
Suppose we have two discrete random variables taking values and resp., and their own probabilities are Denote the joint probabilities Then we have the identities
(1) ( equations).
In words: to obtain the marginal probability of one variable (say, ) sum over the values of the other variable (in this case, ).
The name marginal probabilities is used for 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 and their own densities are and Denote the joint density . Then replacing in (1) sums by integrals and probabilities by densities we get
In words: to obtain one marginal density (say, ) integrate out the other variable (in this case, ).
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.
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 Degrees of Freedom
Table 8 Upper Critical Values of Student’s t Distribution with 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
Open an account at gmail.com, if you haven't already. Open Google Drive.
Install Google sheets on your phone.
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".
Open the file on your drive using Google sheets. Your Calculator is ready!
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.
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.
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).
On the tab named Regression you can calculate the slope and intercept. The sample size must be 10.
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.
Feel free to add new sheets or functions as you may need. You will have to do this on a tablet or computer.
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.
a. Define the density of a random variable 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 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.
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.
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.
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.
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.
You must be logged in to post a comment.