Homework 1
7CCSMRVA - Random Variables and Stochastic Processes
September 28, 2018
Problem 1. If A = {2 ≤ x ≤ 5} and B = {3 ≤ x ≤ 6}, find A ∪ B, AB and (A ∪ B)(AB).
Please note that (AB)c can also be denoted as (AB).
Problem 2. We select at random m objects from a set S of n objects and we denote by
Am the set of the selected objects. Show that the probability that a particular element ζ0 of
S is in Am equals to p = m/n.
Problem 3. A call occurs at time t, where t is a random point in the interval (0, 10).
(a) Find P {6 ≤ t ≤ 8}.
(b) Find P {6 ≤ t ≤ 8|t > 5}.
Problem 4. Box 1 contains 1000 bulbs of which 10% are defective. Box 2 contains 2000
bulbs of which 5% are defective. Two bulbs are picked from a randomly selected box.
(a) Find the probability that both bulbs are defective.
(b) Assuming that both are defective, find the probability that they came from Box 1.
Problem 5. Show that the expected value operator has the following properties.
(a) E[a + bX] = a + bE[X].
(b) E[aX + bY ] = aE[X] + bE[Y ].
(c) Variance of aX + bY = a2 V ar[X] + b2 V ar[Y ] + 2abCovar[X, Y ].
Problem 6. Let p represent the probability of an event A. What is the probability that
(a) the event A occurs at least twice in n independent trials;
(b) the event A occurs at least thrice in n independent trials?
Problem 7. A pair of dice is rolled 50 times. Find the probability of obtaining double six
at least three times.
1
7CCSMRVA - Random Variables and Stochastic Processes
September 28, 2018
Problem 1. If A = {2 ≤ x ≤ 5} and B = {3 ≤ x ≤ 6}, find A ∪ B, AB and (A ∪ B)(AB).
Please note that (AB)c can also be denoted as (AB).
Problem 2. We select at random m objects from a set S of n objects and we denote by
Am the set of the selected objects. Show that the probability that a particular element ζ0 of
S is in Am equals to p = m/n.
Problem 3. A call occurs at time t, where t is a random point in the interval (0, 10).
(a) Find P {6 ≤ t ≤ 8}.
(b) Find P {6 ≤ t ≤ 8|t > 5}.
Problem 4. Box 1 contains 1000 bulbs of which 10% are defective. Box 2 contains 2000
bulbs of which 5% are defective. Two bulbs are picked from a randomly selected box.
(a) Find the probability that both bulbs are defective.
(b) Assuming that both are defective, find the probability that they came from Box 1.
Problem 5. Show that the expected value operator has the following properties.
(a) E[a + bX] = a + bE[X].
(b) E[aX + bY ] = aE[X] + bE[Y ].
(c) Variance of aX + bY = a2 V ar[X] + b2 V ar[Y ] + 2abCovar[X, Y ].
Problem 6. Let p represent the probability of an event A. What is the probability that
(a) the event A occurs at least twice in n independent trials;
(b) the event A occurs at least thrice in n independent trials?
Problem 7. A pair of dice is rolled 50 times. Find the probability of obtaining double six
at least three times.
1
