STAT/MA 416

Books:

1. Introduction to probability, by George Roussas (mandatory)
2. A first course in probability, by Sheldon Ross, 8th edition (optional)
3. A introduction to probability and its applications, by William Feller (optional)
4. Fundamentals of probability: A first course, by Anirban Dasgupta (optional)

Other sources:

(free pdf) Mathematics for computer science (from Chapter 18-Introduction to probability, to Chapter 25-Random Walks), by Eric Lehman and Tom Leighton. Highly recommended.

Lecture Notes 2013

Lecture Notes 2012

Homeworks

Quiz 2+Solutions

Course description

INFORMATION ABOUT THE FINAL

Prepare:
Independence of 2 events, independence of 3 events, pairwise independence of 3 events, relations among them.
Properties of probability, be prepared to apply them in exercises.
Trees and conditional probability, The Product rule, The Total probability formula, Bayes Formula.
How to find F given f, and viceversa, in the continuous 1-dimensional and 2-dimensional cases.
The definition and properties of the distribution function F in the 1-dimensional case.
The definition of the density in both continuous and discrete case. How to find c s.t. f is a density.
The analysis of each type of discrete and continuous random variables. From the discrete ones put the accent on Poisson. From the continuous ones put the accent on the Exponential and the Normal.
The connection between the Poisson and the Exponential (with proof). The memorylessness property of the Exponential. All types of exercises cocerning the Exponential.
The reduction of an arbitrary Normal to the Standard Normal (with proof). All kind of proofs and exercises that require changes of variables involving the Gaussian integral. The value of the Gaussian integral. Exercises using the reduction to the Standard Normal.
The properties of the Gamma function, the definition of the Gamma function. Computations involving changes of variables bringing an integral to the integral from the definition of the Gamma function. The relation between the Gamma function and the Gaussian integral.
Exercises with the Uniform continuous distribution. The probability that a U[a, b] r.v. takes values in a subinterval of [a, b].
Joint density, marginal and conditional densities, etc., all step by step construction leading to the conditional expectation and variance. To carry on all steps of this construction starting from the joint density in both discrete and continuous case.
Properties of conditional expectation, especially the Crucial one. The formula of Total Variance (with ptoof). To know how to apply in exercises the Crucial property of Conditional expectation (like in the problem with the miner). This is the most important chapter.
Independence. Definition, criteria, necessary conditions, proofs, exercises in which we use independence or in which we deduce independence. Accent on criteria 3 and 4. Connection with the m.g.f.
To apply in exercises CLT.

INFORMATION ABOUT MIDTERM 2

Prepare
1)The Hypergeometric distribution.
2)The Uniform distribution both in discrete and continuos case, the differences between the discrete and continuos case like reflected in ex. 2 from Sample Midterm 2, 2010. Exercises similar to those in the lecture notes for the Uniform continuos distribution.
3)The relation between the Poisson distribution and the Exponential one, like reflected in ex. 4 and 6 from Sample Midtm. 2, 2010. How to prove that the waiting time between 2 Poisson occurrances is an Exponential. How to prove Memorylessness and how to apply it.
How to prove that if X is an Exponential, then cX is also an Exponential. How to prove that if X is an Exponential>c, then X-c is an Exponential. How to use that for a Poisson r.v. the parameter is proportional to the length of the subinterval.
4)To use the Var(X) for finding the expectation of X^2. To use the derivatives of the M.g.f. for finding the moments of X.
5)To know the definition of the Gamma function and its properties, to do changes of variables that take you from another integral to the integral in the definition of the Gamma function. The relation between the Gamma function and the Gaussian integral, as in the last ex. from Lecture notes 17. The relation between the Gamma distribution and the Exponential one.
6)To use that if X and Y have the same distribution, then P(X>Y)=1/2. To use that if X is a continuos r.v., then P(X=a)=0 for any real a.
7)To use trees where the conditional probabilities on some branches have to be computed with random variables, like in problems 8 or 9 from the Sample Midtm. 2, 2010.
8)To solve exercises with the Normal using the property that if X~N(a, b^2), then (X-a)/b is a Standard Normal. To do changes of variables that bring an integral to the Gaussian integral. To prove the Crucial Remark from the lesson with the Normal.

STAT/MA 416 sections 041 and 162 , SPRING 2013