KANSREKENING: an introductory course in the basic stochastics (probability and statistics)

Textbook: F.M. Dekking, C. Kraaikamp, H.P. Lopuhaa and L.E. Meester, A Modern Introduction to Probability and Statistics, Springer 2005.

Grading system: homework 30% (earning at least 80% of points is considered as a 100% performance), the midterm test (deeltentamen 1) counts for 30%, and the final exam (deeltentamen 2) for 40%.

  • 09.02.2010. Sections 2.1, 2.2, 2.3, Definition on p. 20, finite sampling models (ordered/unordered with/without replacement).

    Handout and homework problems to be returned 16.02.10.

  • 16.02.2010. The inclusion-exclusion formula, application to the hat-check problem. Section 2.4 complemented with proofs, examples of product probability spaces (coin-tossing, ordered sample with replacement), the binomial distribution (see p. 48), the hypergeometric distribution (see p. 54) appearing in the model of sampling without replacement.

    Handout and homework problems to be returned 23.02.10.

  • 23.02.2010. Sections 3.1, 3.2, 3.3. Tree diagrams and conditional probabilities for multistep experiments, the chain rule of multiplication, application to sampling without replacement.

    Homework problems to be returned 2.03.10:

    1. (1 point) Read Section 1.3 and solve the Monty Hall problem. You may compute as in Problem 3.14, or find your own argument based on conditional probabilities. Generalise to N doors (when moderator opens N-2, or more generally N-k doors).

    2. (1 point) Problem 3.10.

    3. (1 point) Problem 3.11.

    4. (1 point) Problem 3.13.

  • 02.03.2010. Section 3.4 with examples and relation to product measures. Sections 4.1 and 4.2, the properties of distribution functions. Homework problems 3.15, 3.18, 4.2, 4.3 4.6 (1 point each).
  • 09.03.2010. Sections 4.3, 4.4, Poisson distrubution. Uniform distribution on [0,1] as limit of discrete uniform distributions. Section 5.1: distribution functions and densities.

    Homework problems to be returned 23.03.10:

    1. (1.5 points) Suppose for n=1,2,... that a random variable X_n has binomial distribution Bin(n,p). Let n go to \infty and p=p(n) depends on n in such a way that np converges to some positive number \lambda. Show that P(X_n=k) converges to P(X=k) where X has Poisson distribution with parameter \lambda, and k=0,1,... (the case k=0 was treated in the class).

    2. (1 point) Problem 4.11

    3. (1 point) Problem 5.1

    4. (0.5 point) Problem 5.3

  • 23.03.2010. Chapter 5, approximation of the geometric distribution by exponential, properties of the exponential distribution, Section 6.2.

    1. (1 point) Problem 5.7

    2. (1 point) Problem 5.11

    3. (1 point) Calculate the quantile function for the Pareto distribution.

    4. (1 point) Random variables X and Y have Poisson distributions with parameters \lambda and \mu, respectively. Determine the expectation E(2 X + 3 Y).

  • 30.03.2010. Chapter 7; convex functions, Section 8.3; variance of the exponential and normal distributions.

    1. (0.5 point) Problem 7.3

    1. (0.5 point) Problem 7.4

    1. (1 point) Problem 7.7

    1. (1 point) Problem 7.9

    1. (1 point) Problem 7.12

  • Deeltentamen-I is op 20 april 2010 in BBL 001 van 9-12 uur

  • 06.04.2010. Sections 9.1-9.4, pp. 138-139.

    1. (1 point) Problem 9.3

    2. (1 point) Problem 9.12

    3. (1 point) Problem 9.16

    4. (0.5 point) Problem 10.8

  • 13.04.2010. Covariance and correlation, Section 10. The bivariate (correlated) normal distribution.
  • 27.04.2010. Sections 11.1, 11.2. The probability generating function.

    Homework problems:

    1. (1 point) Suppose random variables X_1,...,X_k are independent, and X_j has Poisson(\lambda_j) distribution, j=1,...,k. Find the distribution of the sum S=X_1+...+X_k.

    2. (1 point) Suppose random variables X_1,...,X_k are independent, and each X_j has Geometric(p) distribution. Find the distribution of the sum S=X_1+...+X_k. You may use the generating functions, or any other method.

    3.(1 point) Exercise 11.6 on page 164. Note: the density functions are zero on the negative half-line.

  • 4.05.2010. Chapter 13, frequencies of digits in random number, estimating unknown p in Bernoulli trials, Chapter 14, the central limit theorem, approximation of binomial distribution

    1. (1 point) An aircraft has 300 seats. A passenger who reserved a seat really checks-in for the flight with probability 0.9. The airline management knows this and to avoid losses reserves 324 seats for a particular flight. Estimate the probability of overbooking.

    2. (1 point) Exercise 13.7

    3. (1 point) Exercise 13.12.

  • 11.05.2010 The maximum likelihood method, estimator for N in the hypergeometric distribution, Sections 16.1-2-3, 19.1,19.3, 19.4, 20.1-2

    1. (1 point) Exercise 19.2 questions (a), (b) from the textbook. One more question (c): assuming that the variables are i.i.d. find the unbiased estimator T of the form as in (b) for which Var(T) is minimal.

    2. (1 point) For a sequence of n i.i.d. Bernoulli trials with sucess probability p, derive the maximum likelihood estimator T for p; calculate the bias of T and Var(T).

    3. (1 bonus point) Exercise 20.12.

    Comment: bonus points are granted for exercises not included in the main set. The bonus points may improve nevertheless your homework scores.

  • 18.05.2010: p. 305, proof of the Cramer-Rao bound, Fisher's information in Bernoulli trials, pp. 317-318, pp 320-321, pp 345-346

    (1 point) Let X_1,...,X_n be a i.i.d. sample from the Poisson distribution with unknown parameter \lambda. (a) Derive the Cramer-Rao bound for the variance of any unbiased estimator of \lambda. (b) Compare the bound with the mean-square error of the natural estimator T=(X_1+...+X_n)/n.

    (1 point) Exercise 20.10

    (1 point) Exercise 21.6

  • 01.06.2010: pp 348-350, section 24.2, pp. 329-331.

    (1 point) 22.9

    (1 point) 23.3

    (1 point) 23.10