Lecture 1 30/8 Ch 2: events, probabilities, Kolmogorov's axioms, Venn diagrams, some combinatorics Recommended problems for private study: p8, p27 and p33 in selected exercises plus this one: Let A and B be two events such that P(A)=0.6, P(B)=0.7 and P(A∪B)=0.8. Compute P(A∩B). (You can also choose among the problems at the end of chpter 2 in the textbook.) Lecture 2 31/8 Ch 2: statistical independence, conditional probabilities Recommended problems for private study: p1, p13, p32 and p34a in selected exercises Lecture 3 1/9 Ch 2: law of total probability, Baye's rule Recommended problems for private study: p11 and p34b in selected exercises plus these two: 1. 65% of all email to an account is spam. The spam filter, which is not totally perfect, classifies 60% of all incoming mail as spam, and puts them in a spam folder. It turns out that 96% of the mails in the spam folder is in fact spam, whereas 4% are not. Calculate the probability that the filter correctly classifies a spam mail as such. 2. (Exam question 1b Aug. 17 2015) In a certain region 20% of the population are smokers. The probability for a smoker to get lung cancer is ten times as high as for a non-smoker. The probability that a randomly chosen person gets lung cancer is 0.006. Compute the probability that a person who smoke gets lung cancer. Look also at the very first problem on each of the exams I have linked to on the main page. Lecture 4 6/9 Ch 3, 6, 9: discrete random variables, expectation and variance for such Exercises here. (I think I will have time to do much of lecture 5.) Lecture 5 7/9 Ch 3, 6: continuous random variables, expectation and variance for such Exercises here. Lecture 6 12/9 Ch 4, 5: law of total probability for continuous distributions, max and min of random variables, the convolution formula for sums Exercises for private study: 1. X is Bin(7, 0.4) and Y is Bin(6, 0.3) and they are independent. Compute P(X+Y = 4). (Ans: 0.2211) 2. X and Y are exp(2) and exp(3), respectively (with the current notation, not the textbook's) and independent. Let Z = Max(X, Y) and compute the expected value E(Z). (Ans: 19/30) 3. X and Y have density functions fX(x) = x e-x, x ≥ 0 fY(y) = 4y e-2y, y ≥ 0 Compute the probability P(X > Y) (Ans: 20/27) Lecture 7 14/9 Ch 7: general properties for expectations and variance, covariance and correlation Exercises: 1. X and Y are r.v. such that V(X) = 2, V(Y) = 3 and V(X+Y) = 3. Compute C(X, Y) (Ans: –1) 2. X ∈ Bin(10, 0.1) and Y ∈ Bin(5, 0.1) are independent. Let Z = X + 2Y. Compute a) E(Z) and V(Y) b) P(Z = 3) (Ans: a: 2 and 2.7;   b: 0.1610) 3. The independent r.v. X and Y have the density functions fX (x) = x/2,   0 ≤ x ≤ 2;   fY (y) = 6y(1 – y),   0 ≤ y ≤ 1. Compute the correlation coefficient ρ(X + Y, Y). (Ans: 3/7) Lecture 8 20/9 Ch 8: the normal distribution, Central Limit Theorem Lecture 9 22/9 Ch 12: point estimates of parameters, unbiased estimates Exercises: 1. x1, ..... ,xn are independent observations of  Po(λ). Prove that the ML-estimate of λ is the mean value of x1, ..... ,xn. 2. Recall exercise 9.3 in exercises. a) Compute the LS estimate of α. Note that it differs from the ML-estimate. b) Prove that E(X2) = 3α c) Compute the MK-estimate of 3α using the result in b). Compare with the ML estimate of α. Lecture 10 27/9 Ch 13: confidence intervals Exercises: 1.  S.3.a in selected exercises 2.   S.20 in selected exercises Use a one sided confidence interval a  ≤  μA-B  for μA-B. Lecture 11 30/9 Ch 13: more on confidence intervals Exercises: S.13.b, S.17 and S.28 in selected exercises. Lecture 12 3/10 Ch 14: hypothesis testing. Exercises: 1. Joe Doe wants to test the hypothesis λ = 40, where λ is the intensity of a poisson distribution, Po(λ). He will collect one observation of X ∈ Po(λ) and reject the hypothesis if X ≤ 28 or if X ≥ 54. Compute the exact (3 sign. digits) error risk for this test. (Ans: 4.94%) 2. Like in the previous example Joe Doe wants to test the hypothesis λ = 40, where λ is the intensity of a poisson distribution, Po(λ). He will collect one observation of X ∈ Po(λ) and reject the hypothesis if the confinence interval for λ does not contain 40. He employs the approximate "Z-interval" (i.e., normal approximation) at the approximate 95% confidence level. What is the exact (4 sign. digits) error risk for this test? (Ans.: 5.725%.) Lecture 13 7/10 Ch 14: more on hypothesis testing, power of a test, power function Lecture 14 11/10 Ch 14: χ2-tests Lecture 15 13/10 recap and catch up. I leave regression analysis for private study.