Week 11 | Moments of multivariate distributions
Textbook sections: 4.64.8
Keywords:
Product moments, linear combinations of random variables, conditional expectations.
Exercises
11.1. Read textbook example 4.15. Suppose that the discrete random variables
Week 8 | Special distributions: The binomial models
Textbook sections: 5.15.5, 5.8, (5.6, 5.9).
Keywords:
Bernoulli trials, binomial, geometric, negative binomial distributions, multinomial distributions.
Exercises
8.1. Find the mean of the random variabl
Week 10 | Special distributions: The normal and related distributions
Textbook sections: 6.56.7
Keywords:
Normal and bivariate normal distributions.
Exercises
10.1. Read textbook examples 6.26.4, and evaluate the following quantities.
(a) For Z N (0, 1);
Week 7 | Some special expectations & Midterm exam
Textbook sections: 4.5
Keywords:
Moment generating function, cumulant generating function.
Taylor series: Let f be a smooth function on some interval I R.
The Taylor series for f at a I is
f (x) = f (a) +
Week 3 | Bayes Theorem and random variables
Textbook sections: 2.6-2.8, 3.1-3.2
Keywords:
The law of total probability, Bayes theorem, random variable and its distribution.
More tips for computing probabilities: Suppose we have a (random) sequence consist
Week 9 | Special distributions: The Poisson and related models
Textbook sections: 5.7, 6.26.3
Keywords:
Poisson, gamma, exponential and chi-square distributions
Exercises
9.1. Let X Binomial(n, /n) for some > 0 and a large natural number n.
(a) By using t
Week 4 | Probability distributions
Textbook sections: 3.23.4
Keywords:
Probability mass function, probability density function, (cumulative) distribution function.
Integrals:
Some integrals: For u := u(x), v := v(x),
Z
Z
Z
Z
Z
Z
1
cudx = c udx,
(u + v)dx
Week 5 | Probability distributions of two or more random variables
Textbook sections: 3.53.7
Keywords: Joint pdf, joint distribution function, marginal and conditional distributions, independence between two or more random variables.
Exercises
5.1. (Durre
Week 2 | Mathematical introduction to probability
Textbook sections: 2.22.7
Keywords:
The axioms of probability, disjoint and independent events, conditional probability.
A few tips for writing probabilities: Suppose we have a sequence consisting of peopl
Week 12 | Distributions of functions of random variables I
Textbook sections: 7.27.4, 6.4.
Keywords:
Distribution function technique, change-of-variable (transformation) technique.
General strategy (X Y ): Let X be a random variable with pdf (or pmf) fX o
Week 1 | Probability by counting
Textbook sections: 1.11.3, 2.12.2
Keywords: Sample space, outcomes and events, probability by counting, basic principle of counting, treediagram, arrangement of objects, partition of objects, binomial coefficient.
For this
Week 14 | Concentration of distributions
Textbook sections: 4.4, 6.6, 8.18.2
Keywords: Law of large numbers and central limit theorems, normal approximation of binomial distribution,
Chebyshevs theorem.
Markovs inequality:
, then for any c > 0,
Let X be a
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by Ca
Week 13 | Distributions of functions of random variables II
Textbook sections: 7.5
Keywords:
The m.g.f. technique.
Exercises
13.1. (Do not submit) Read textbook Theorem 7.3, and examples 7.157.16.
13.2. Using the m.g.f. technique, show the following:
(a)