KTH Mathematics / Mathematical Statistics   

Current Information for TCOMK3 autumn 2016


Wednesday 26/10
the exam with solutions (in Swedish).

Thursday 13/10
I demonstrated this old exam.

Friday 7/10
Chi-square tests.

Here are some old exams in English:

Friday 7/10
– Normal approximation of Binomial and Poisson distributions

– confidence intervals for p Bin(n,p) and difference of such

– confidence intervals for λ in Po(λ) and difference of such

– some examples on power of hypothesis tests

– the p-value method.

– the compilation of formulae, the numeric tables and the quick reference for the TI82-STATS are now in English.

Monday 3/10
I described how to compute one-sided confidence intervals (C.I.) on TI82

Hypothesis testing using C.I., one-sided and two-sided.

"Direct" method to construct tests with given error rate. We noted that when the distribution of interest is discrete, we have only a discrete choise of error levels. In the first example, we wanted an error level of 10%, but we had to choose either 10.34% or 4.80%. (We took the nearest, 10.34%.)

By the same token, there are no CI:s with exact confidence levels for discrete distributions.

I talked a little on The p-value fallacy; If we reject the null, the error rate ("p-value") is NOT the probability that the null is true. This is however whata majority of those who believe they understand hypothesis testing (falsely) believe.

Finally, I talked a little on the power of a test, and we compared the power of two tests with the same error risk, and found that one was more powerful than the other, and hence the test to choose.

Exercises for you in private as usual in the plan for lectures.

Friday 30/9
I was ill, so Salah took both lectures and exercises.
I am back Monday.

Exercises for you in private as usual in the plan for lectures.

Tuesday 27/9
There are exercises for you in the plan for lectures. As for the second exercise: A one sided confidence interval with confidence 95% may be constructed like this: Compute the two sided interval (as we did in the lecture,) but with confidence level 90% (double the risk error). Then just employ one of the sides, like a ≤ μ.

Confidence intervals for

1. mean value of a normal distribution, σ known ("ZInterval")

2. d:o σ unknown ("TInterval")

3. approximate interval when we have many observations, not necessarily normal distribution, CLT ("ZInterval")

4. difference in mean values of normal distributions, σ:s known ("2-SampleZInterval")

5. d:o when σ:s unknown but assumed equal ("2-SampleTInterval, pooled=YES")

(6. d:o when σ:s unknown but not assumed equal ("2-SampleTInterval, pooled=NO")

7. approximate difference in mean values of a distributions, not necessarily normal distributions, unequal variances, (CLT) ("2-SampleZInterval").

8. Pairwise observations

Thursday 22/9
There are two exercises for you in the plan for lectures.

I demonstrated LS and ML point estimators. In particular, I showed that the LS estimate of a parameter is the same as the first moment estimator. I used this to estimate the intensity of an exponential distribution.

As for ML, I took a small example of a discrete variable, the exponential distribution, the binomial distribution and the general case Po(tiλ), i=1....n.

Tuesday 20/9
I reminded of some of the rules for expectation, variance and covariance. Then I talked about the normal distribution; it's different features and the Central Limit Theorem. I illustrated with two examples.

New Examiner
The examiner is now Thomas Önskog, previously it was Tatjana Pavlenko. Not that it matters much, but if you are interested in old exams, here are the only ones I could find made by Thomas.

Jan 2016
June 2016

(It is a fact that different examiners put somewhat different emphasis on different elements of the course.)

Quiz Exam 19/9
The information in Swedish I translated says Friday 19/9, but there is no such day. It is Monday 19/9. The rooms are these:

L21, L42, L43
M3, M24, M35, M36, M37, M38
Q11, Q15, Q17, Q22, Q24, Q26, Q31, Q33
V01, V34

and the time is 15 &ndash 17. Bring your calculator, but no compilation of formulae, etc.

You should have got a mail about this. If you have any questions, turn to the student affairs office

Wednesday 14/9
I talked on the rules for calculating expected values (linearity) and introduced covariance. I showed the rules for covariances (bilinearity) and said that the easiest way to handle variance of a r.v. is to treat it as the covariance of the r.v. with itself.

Then I defined correlation coefficient, and we looked at some examples.

Note that the lectures up to and including this one cover the stuff treated on the quiz exam Sept. 19.

You find exercises in the lecture plan.

Monday 12/9
The instructions for the first bonus-workshop is now available in inglish

I talked about multivariate random variables, in particular two dimensional discrete r.v. and the convolution formula for sums. I also derived formulae for the probability function for the maximum and minimum of two discrete r.v.

Next I did the corresponding for continuous variables: the convolution formula for sums, and the densities for maximum and minimum of two continuous r.v.

I also demonstrated the continuous version of the law of total probability, and used it to to compute P(X > Y) when X and Y are two independent continuous r.v. In particular, I made the calculation for X ∈ exp(λ) and X ∈ exp(μ).

Exercises in the lecture plan.

Friday 9/9
The instructions for the first bonus-workshop is now available in inglish

Wednesday 7/9
I lectured according to the plan. Idemonstrated the syntax in TI-82 STATS to calculate a probability P(aXb) when X is a continuous r.v.

I defined the distribution function for a continuous r.v. and its properties.

Next I derived a general formula for the density of a function of a continuous r.v., i.e., if the density of the r.v. X is known, and Y = g(X, whow we calculate the density for Y. I gave a number of examples.

As usual, You find recommended exercises for private study in the plan for lectures or directly here.

Monday 5/9
I lectured according to the plan, and demonstrated useful features in TI-82 STATS (there ase similar features in other makes,) and also started with continuous r.v. I introduced the exponential distribution (note that the English edition of Blom uses the notation exp(1/λ) whereas we on exams etc. use exp(λ) if the density function is f(x) = λe-λx.)

I pointed out the relationship between the Poisson and exponential distribution.

I used the notation SD(X) for standard deviation, whereas I see now that both Blom and "we" in exams etc. employ the notation D(X) for the standard deviation. I will stick to SD(X), since I am so used to it that I don't think I can manage to consistently use D(X). So please note that D(X) and SD(X) both denote standard deviation.

Read about the binomial and poisson distributions in ch. 9 (7 in the Swedish ed), sections a) and b), i.e. "occurrence" and "exact properties".

As usual, You find recommended exercises for private study in the plan for lectures or directly here.

Thursday 1/9
Here is information on the exam, the quiz, and the computer workshops. It is essentially a Google translation of the Swedish web page. I hope it is intelligible. The instructions for the introductory workshop on Monday is now in English. Camilla Landén is working on a translation of the other.

I lectured according to the plan, and also started with the FFT, binomial and hyper geometric distributions and showed how they are implemented on a TI82 Stats.

I made a punch error, it should be

binompdf(20, 0.2, 5) = 0.1746.

Again,You find recommended exercises for private study in the plan for lectures

Wednesday 31/8
I lectured according to the plan, and as promised i proved the formula

P(A) = P(A|B)P(B) + P(A|B*)P(B*).

and gave some examples.

I solved problem 1a) in the Aug.2014 exam, and proved that if the events A and B are independent, then also A and B* are independent etc. (exercise 220 in Blom.) I also demonstrated an easier version of exercise 218 and one somewhat similar to the Monty Hall problem.

Again,You find recommended exercises for private study in the plan for lectures

Tuesday 30/8
You find recommended exercises for private study in the plan for lectures

Note that you must register for the course in Rapp. Login with your KTH-id and click "activate". The code for this course is "sanstat16".

There will be a quiz exam sept. 19 at 15:00 – 17:00. It is not compulsory, but will give some bonus on the final exm. More information later.

I lectured according to the plan. I had some difficulties to convince some of the students that if you toss a coin twice, the probability to get heds both times is 1/4. They insisted that the probability is 1/3. (Three possibilities, two heads, one head one tail, two tails). Actually, I don't think I managed.

Anyway, tomorrow it's conditional probabilities. Salah: I will prove the this simple version of the "law of total probability":

P(A) = P(A|B)P(B) + P(A|B*)P(B*).

So you can employ this on the "MALMO" problem.

Wednesday 10/8
I have now updated the exercises pdf, I hope all is correct now.

Sidansvarig: Harald Lang
Uppdaterad: 2016-08-14