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Math351 Spring 2010
Final review
Warning: this is rather a review of the course than a review before the ﬁnal
exam. Consider it as a supplement to the excellent endofchapter reviews in the
Ross’s book. This review consists of two parts:
(1) A short list of basic notions (and only some facts). Make sure it is clear to
you “what is what”. Typically, it is possible to deﬁne each of these notions in one
clear sentence.
1
Try to do so. Only sometimes you will need to write a formula.
(2) A collection of “naive” questions. Most questions can be answered very
quickly, without any calculations or without long calculations. These questions are
not, in general, a model for the ﬁnal exam problems. Most questions do not contain
traps; they do not seem
easy, they are
easy.
1.
Basic notions
•
Sample space
•
Outcome
•
Event
•
Mutually disjoint events
•
Probability of an event
•
Independent events
•
Conditional probability
•
Reduced sample space
•
“Conditioning”
•
Bayes’s formula
•
Random variable
•
The indicator variable of an event
•
Cumulative distribution function of a random variable
•
Discrete random variable
•
Continuous random variable
•
Probability mass function of a discrete random variable
•
The density function of a continuous random variable
•
Expectation of a random variable (in the discrete case, and in the contin
uous case)
•
Variance of a random variable
•
Standard deviation of a random variable
•
The hazard rate function (associated with a random variable)
•
Joint cumulative distribution function of two random variables
•
Joint probability mass function of two discrete random variables
•
Jointly continuous random variables, joint density function
•
Independent random variables
•
Conditional probability mass function
•
Conditional probability density function
•
Change of variables in distributions and joint distributions
•
Covariance of two random variables
•
The correlation coeﬃcient of two random variables
•
Positively, negatively, zero correlated variables
1
Except for the most basic notions: sample space and probability
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Conditional expectation
•
Moment generating function of a random variable
•
Multiplication rule for moment generating functions of sums of independent
random variables
•
Uniqueness rule for moment generating functions
•
Markov’s inequality
•
Chebyshev’s inequality
2.
Sample space, events, probabilities.
..
(1) What is
P
{
2 + 2 = 5
}
?
(2) Suppose
P
{
A
}
= 0. Does this mean that
A
is impossible?
(3) Suppose
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This note was uploaded on 03/05/2011 for the course MATH 351 taught by Professor Moumen,f during the Spring '08 term at George Mason.
 Spring '08
 Moumen,F
 Math, Probability

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