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

f_X(x) = x e^-x, x ≥ 0
f_Y(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

f_X (x) = x/2,   0 ≤ x ≤ 2;   f_Y (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, ..... ,x_n are independent observations of  Po(λ). Prove that the ML-estimate of λ is the mean value of x₁, ..... ,x_n.

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  ≤  μ_A-B  for μ_A-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.