First course in probability and statistics
Department of mathematics and systems analysis
Aalto University
3B
J Tölle & S Moradi
Spring 2017
Exercise 3B
Continuous random numbers
Cumulative distribution function
Normal distribution
Stochastic dependence and linear correlation
Class exercises
3B1 Consider the random variable Z ∼ N (0, 1).1
(a) Find the median of Z, that is, a number z such that Pr(Z > z) = 0.5.
(b) What is Pr(Z > 1)?
(c) What is Pr(Z ≤ −1)?
(d) Find z such that Pr(Z ≤ z) = 0.95.
(e) Find z such that Pr(Z ≥ z) = 0.05.
(f) What is Pr(|Z| ≤ 2)?
(g) Find z such that Pr(|Z| ≥ z) = 0.05.
Consider the random variable X ∼ N (1, 9).
(h) What is Pr(X ≤ −1)?
(i) Find x such that Pr(X ≥ x) = 0.05.
(j) What is Pr(X = 1)?
3B2 (The correlation of supporter groups) In the upcoming elections, the relative votes for
the two largest parties are modeled by two random numbers X and Y , which have the
joint distribution given by the following density function
(
2,
x, y ≥ 0, x + y ≤ 1,
f (x, y) =
0,
otherwise.
(a) Determine the density function, the expectation and the standard deviation for X.
(b) Determine the density function, the expectation and the standard deviation for Y .
(c) Compute the correlation of X.
(d) Determine, whether X and Y are dependent or independent.
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Consider Mellin’s statistical tables, too. See the following MyCourses page:
https://mycourses.aalto.fi/mod/resource/view.php?id=197649.
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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 3B
Homework
3B3 Suppose that random variables X and Y have joint density function
f (x, y) = Cxy,
0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
where C is a constant. Determine::
(a) The value of C.
(b) Pr(0 ≤ X ≤ 1/2, 1/2 ≤ Y ≤ 1)
(c) Density functions of X and Y .
(d) Are X and Y dependent or independent?
3B4 Suppose that the probability mass function of random vector (X, Y ) is
Pr(X = 2, Y = 4) = Pr(X = −2, Y = 4) = Pr(X = 0, Y = 0) = 1/3.
(a) Calculate the covariance between X and Y .
(b) Are X and Y linearly correlated?
(c) Are X and Y dependent?
Explain your results.
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A fast algorithm for tracking human faces based on

Mutation rate