First course in probability and statistics
Department of mathematics and systems analysis
Aalto University
2B
J Tölle & S Moradi
Spring 2017
Exercise 2B
Random variables and probability distributions
Cumulative distribution function
Statistics of distributions
Discrete distributions
Continuous distributions
Class exercises
2B1 A gambler gambles on a game where he either wins or loses one euro on each round.
In the beginning, the gambler has one euro and the probability of win on each round is
1/4. The gambler decides to play exactly three rounds. When there are no more euros,
the gambler can’t play anymore. Let us define a random variable by setting
X = the gambler’s cash account when he quits gambling.
(a) What are the probabilities of events X = 0, 1, 2, 3, 4 and what is the probability
mass function of X? Draw a graph of this function.
(b) What is the cumulative distribution function of random variable X? Draw the
graph of this function.
(c) What is the probability of event X = 1.5?
(d) Find the probability of event X > 1 using the probability mass function.
(e) Find the probability of event X > 1 using the cumulative distribution function.
2B2 The density function of a random variable X is given by
(
ax2 − 2ax,
if 0 ≤ x ≤ 1
f (x) =
0,
otherwise.
(a) What is the value of a?
(b) What is the cumulative distribution function of X?
(c) What is the probability of event X = 0.5?
(d) Find the probability of event 0 ≤ X ≤ 0.25 using the probability mass function.
(e) Find the probability of event 0 ≤ X ≤ 0.25 using the cumulative distribution
function.
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First course in probability and statistics
Department of mathematics and systems analysis
Aalto University
J Tölle & S Moradi
Spring 2017
Exercise 2B
Homework
2B3 (A model for salary distribution) We model the salary (EUR) of a randomly chosen
employee as a random variable X, with density function
(
αcα x−α−1 ,
x > c,
f (x) =
0,
otherwise,
where α = 1.6 ja c = 1300.
(a) Compute the cumulative distribution function of X and draw a picture of it.
(b) Determine the value set of X, that is, how small and how large values can X take?
(c) Compute the probability that some randomly chosen employee earns more than
13 000 EUR/month.
(d) Determine a salary level z, such that 90% of employees earn less than z EUR in
one month.
2B4 A person is taking a test, where she types on a computer until the first typo occurs. The
number of characters written during this test is a discrete random variable X. Suppose
that the probability of a typo for each character is p, 0 < p < 1 and these events are
independent. What is the probability mass function of the random variable X? What
is the expectation of the distribution?
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