Download Question 1 Illustrate the concept of probability space. In particular: (a

Professor Paolo Vitale
Revision Questions: Elements of Probability
Econometrics
Academic Year 2016-2017
Question 1
Illustrate the concept of probability space. In particular:
(a) define its elements;
(b) show the properties of a sigma field;
(c) and those of a probability function.
Question 2
(i) Define a random variable.
(ii) Explain how a probability space on the random variable is derived from the probability space on the
set of states of nature.
Question 3
Define the cumulative distribution function associated with a random variable and its properties.
Question 4
1. Clarify the difference between discrete and absolute continuous random variables.
2. Describe how their cumulative distribution functions differentiate.
3. Propose some examples of discrete and absolute continuous random variables.
Question 5
(a) Define the mean and the variance for discrete and absolute continuous random variable.
(b) Explain what they represent.
(c) Discuss the main properties of the mean and the variance.
Question 6
(i) Define skewness and kurtosis.
(ii) Explain the information these two moments provide on the distribution of a random variable.
Revision Questions
Econometrics
Question 7
1. Define the covariance and the correlation between two random variables.
2. Clarify the information they provide.
3. Discuss their properties.
Question 8
(a) Define the notion of conditional probability.
(b) Explain when two events are independent and clarify why this concept is symmetric.
Question 9
(i) Define the conditional cumulative distribution function for random variables.
(ii) Explain how the conditional distribution can be identified from the joint distribution in the case of
discrete and absolute continuous random variables.
Question 10
Define the conditional mean and variance of discrete and absolute continuous random variables.
Question 11
1. Define the notion of independence for two random variables.
2. Discuss the properties of their marginal distributions.
3. Discuss their properties of the conditional moments.
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