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ECON 2070
PROBABILITY REVIEW EXERCISE
This is an optional exercise. There is no need to turn it in. However, I strongly
encourage you to try this exercise out if
• you feel that you need a brush up on your probability theory;
• you are unsure whether your knowledge on probability is sufficient for this course;
or
• you have nothing better to do.
The solution to this exercise is available on Blackboard.
1. Probability and Conditional Probability
1. Suppose we toss 3 fair coins (i.e., each coin has 1/2 chance of landing on head).
The tosses are independent.
(a) Write down the outcome space and the probability of each outcome.
(b) What is the probability that the first coin comes up head?
(c) What is the probability that at least two of the three coins come up heads?
(d) What is the probability that, somewhere in the sequence (anywhere will do),
there are two consecutive heads?
(e) Conditional on the first coin coming up head, what is the probability that
at least two of the three coins come up heads?
(f) Conditional on the first coin coming up head, what is the probability that
there are two consecutive heads?
2. Expectation and Conditional Expectation
1. Find the expectation of the random variable X which takes the value
5 with probability 0.2
8 with probability 0.7
12 with probability 0.1.
1
2. Consider a random variable X which follows the uniform distribution on [a, b].
You may note that the cdf of such random variable is
F (x) =
0 if x < a
x−a
b−a if a ≤ x ≤ b
1 if x > b
;
and the pdf of such random variable is
f(x) =
1b−a if a ≤ x ≤ b0 otherwise.
(a) Find the expectation of X.
(b) Let c be a number such that a < c < b. Find the conditional expectation of
X given that X is less than or equal to c.
3. A new technology in producing widgets is available. However, as the technology
is still relatively new, the number of widgets that can be produced using this
technology is random. In particular, the output will be
1 with probability 0.3
4 with probability 0.4
9 with probability 0.3.
(a) What is the expected number of widgets produced?
(b) Consider Firm 1 whose profit function is as follows: if it produces x widgets,
it gets a profit of
√
x dollars.
(i) What is the expected profit of Firm 1?
(ii) Is the expected profit of Firm 1 higher than, equal to, or smaller than
its profit at the expected number of widgets?
(c) Now consider Firm 2 whose profit function is as follows: if it produces x
widgets, it gets a profit of x2 dollars.
(i) What is the expected profit of Firm 2?
(ii) Is the expected profit of Firm 2 higher than, equal to, or smaller than
its profit at the expected number of widgets?