Math 362 - Review #3 - Spring 2003
Introduction
to Probability Theory
Review for Exam # 3
Close all books and notebooks and ask yourselves the following questions.
- Bivariate Distributions
- What is the joint probability function of a pair of discrete
random variables?
What kind of information does it give you?
What are its properties? Give an example.
- How would you compute Pr{X in A,
Y in B} for discrete random variables X
and Y
- How would you compute the marginal probability functions from the
joint probability function.
- Using the joint probability function, how would you determine if
two discrete random variables X and Y are
independent.
- What is the joint probability density function of a pair of continuous
random variable? What kind
of information does it give you? What are its properties? Give an example.
- How would you compute Pr{a < X < b,
c < Y < d} for continuous random variables X
and Y
- How would you compute the marginal density functions from the
joint density function.
- Using the joint probability function, how would you determine if
two continuous random variables X and Y are
independent.
- What is an identity for
E[g(X)h(Y)] in the case that
X and Y are independent random variables?
- How would you be able to tell whether a given function could or could not
be
- a joint probability function?
- a joint probability density function?
- Conditional Distributions
- What is the formula for fY|X
(y|x) for discrete random variables?
- Give the step to compute Pr{X=x|Y=y}
for discrete random variables X and Y.
- What is the formula for fY|X
(y|x) for continuous random variables?
- Give the step to compute Pr{X < x|Y=1}
for continuous random variables X and Y.
- Give the step to compute Pr{X < x|Y=y}
for continuous random variables X and Y.
- Functions of a Random Variable
- If X is a continuous random variable and if
Y = g(X) for a monotone function g, what is
the density of Y given the density of X?
- What is the probability transform?
- Why is it useful?
- Variance and Covariance
- Give the basic definitions of variance and covariance.
- Give alternative expressions for variance and covariance.
- Explain what it means to standardize a random variable.
- Define correlation.
- What values can the correlation take?
- What does a correlation of -1, -0.7, 0.2. 0.8, and 1 mean?
- Show that if two random variables are independent, then they are
uncorrelated.
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