Download Probability and Statistics Set 1

Probability and Statistics
Set 1
1. Find
S∞
i=1 Ai
and
T∞
i=1 Ai ,
if Ai are defined in the following way:
1
(a) Ai = {x ∈ R : 0 ≤ x ≤ (i+1)
},
√ √
(b) Ai = x∈ R : i ≤ x ≤ 2i
n
1
(c) Ai = x ∈ R : 1 − i+1
≤x≤2+
1
i+1
o
(d) Ai = {x ∈ R : sin x = i}.
2. Find counterimages f −1 ((−∞, a >) and f −1 (a), where: a ∈ R and function f is defined in the
following way:
(a) f : R → R, f (x) = |x − 12 |

 0 for
1 for
(b) f : R → R, f (x) =

2 for

 x + x0
0
(c) f : R → R, f (x) =

x − x0
x ∈< 0; 81 > ∪ < 38 ; 48 > ∪ < 68 ; 87 >
x ∈ ( 18 ; 28 > ∪( 48 ; 85 > ∪( 78 ; 1 >
x ∈ ( 28 ; 38 ) ∪ ( 56 ; 68 )
for x ∈ (−∞; −x0 )
for x ∈< −x0 ; x0 >
for x ∈ (x0 ; ∞)
(d) f : {0.1, ..., 2k} −→ R, f (x) = |x − k|.
x − y for x ≥ y
(e) f :< −1, 2 > × < −1, 2 >−→ R, f (x, y) =
0
for x < y
2
2
x + y for x ≥ y
(f) f : R2 → R, f (x, y) =
0
for x < y
(g) f : {−1, 0.1, 2} × {−1, 0.1, 2} −→ R, f (x, y) = max(x, y) − min(x, y).
3. Among the chairs in a given classroom 60% are green and 70% are made of wood. How many
non-green wooden chairs are there
(a) if we assume that the colour of a chair and the material it is made of are independent?
(b) if we do not assume that?
4. A die (traditional, with numbers 1,...,6) is tossed 5 times. Calculate the probability that the
greatest number observed is:
(a) equal to 4
(b) less than 4.
5. Seventy percent of all vehicles examined at a certain technical inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate
the following probabilities:
(a) P(all of the next three vehicles inspected pass)
(b) P(at least one of the next three inspected fail)
(c) P(exactly one of the next three inspected passes)
(d) P(at most one of the next three vehicles inspected passes).
6. A fair coin is tossed 10 times. Compute the probability that the number of heads obtained:
(a) equals 0
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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Probability and Statistics
(b) equals 1
(c) equals 5
(d) is even
7. A fair coin is tossed 2008 times. What is the probability that the number of heads obtained is
even?
8. A real estate agent is showing houses to a prospective buyer. There are ten suitable houses in
the area but the buyer only has time to visit four of them.
(a) In how many ways can the four houses be chosen if the order of visiting is considered?
(b) In how many ways can they be chosen if the order is disregarded?
(c) If four of the ten houses considered are newly built and six have previously been occupied,
what is the probability that all four houses visited by the buyer are new? Does the answer
depend on whether the order is considered?
9. Four people (N, E, S, and W) are playing bridge, i.e. each of them has 13 cards of a 52-card
deck. Assume that N and S have 8 spades in total. Compute the probability that
(a) E has exactly 3 spades
(b) E has exactly 2 spades
(c) E has no spades.
10. Two engines of a ship fail independently of each other. The probability that only the older
engine will fail on any given day is 0.15, and the probability that only the newer one will fail is
0.05. What is the probability that both engines will fail on a given day?
11. Eight volumes of an encyclopedia are put on a shelf in a random order. What is the probability
that none of them occupies its proper place?
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ECE, 2008/2009, winter semester
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Probability and Statistics
Set 2
1. We throw two dice and compute the sum and the product of the results. Consider the following
events: A - the sum is even, B - the sum is divisible by 3, C - the product is even, D - the
product is divisible by 3. Compute the probabilities of those events and decide which are pairwise
independent.
2. There are 10 balls in the urn: 5 large, 3 medium-sized and 2 small ones. We take two of them
at random. Calculate the probability that:
- they are the same size;
- at least one of them is large;
Consider separately two cases:
(a) the first ball is returned to the urn
(b) it is not.
3. Two points belonging to the interval [0; 1] are chosen at random. The interval is thus divided
into three smaller ones. What is the probability that:
(a) the leftmost subinterval is the longest;
(b) the rightmost subinterval is the shortest.
Are the above events independent?
4. Three points A, B, C were chosen at random on the circumference of a circle O. What is the
probability that the angle ABC is:
(a) acute
(b) right
(c) obtuse?
5. There are three major manufacturers of whistles: A, B and C, with market shares 0.5, 0.3 and
0.2. 4% of A’s whistles are faulty and so are 6% of B’s and 5% of C’s. We buy a random whistle.
(a) What is the probability that it works properly?
(b) What is the probability that was made by A provided that it works?
(c) What is the probability that was made by A provided that it is out of order?
6. In the setting of Ex. 2 compute the (conditional) probability that:
(a) at least one of the balls is large provided that they are the same size
(b) the larger one is medium-sized provided that they are not the same size.
7. In the setting of Ex. 3 compute the (conditional) probability that:
(a) the leftmost interval is > 1/2 provided that it is the longest
(b) the leftmost interval is < 1/2 provided that it is the longest
(c) the leftmost interval is < 1/2 provided that it is not the longest
(d) the leftmost interval is not the longest provided that it is < 1/2.
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Probability and Statistics
Set 3
1. A coin is tossed n times. Let A denote the event ’all results are the same (i.e. n heads or n
tails)’, and let B mean ’the number of tails is not greater than 1’. For what values of n are A
and B independent?
2. We choose a family with two children. What is the probability that both children are boys if we
know that the older child is a boy? What is the probability that both children are boys if we
know that at least one child is a boy?
3. There are 10 white and 10 black balls and two urns. The player has to do the following:
- place the balls in the urns (arbitrarily, but no urn can remain empty),
- toss a coin,
- draw a ball from urn 1 if the coin landed head-up, from urn 2 otherwise.
The player wins if the ball is white. What distribution of the balls among the urns maximizes
the probability of obtaining a white ball?
4. A and B toss a coin in turns (A begins). The first player to obtain a head wins. What is the
probability that A will win?
5. A and B toss a die in turns (A begins). The first player to obtain a ’6’ wins. What is the
probability that A will win?
6. A, B and C toss a coin cyclically (A → B → C → A → ...). The first player to obtain a head
wins. What is the probability that B will win?
7. A, B and C toss a coin in the following order: (A → B → C → B → A → B → C → ...). The
first player to obtain a head wins. What is the probability that B will win?
8. The banks A, B and C have the following numbers of clients: A - 12 million, B - 4 million and C
- 8 million. According to a recent poll, some of the clients are ’very satisfied’ with the quality of
service - the shares are 20% for A, % for B, and 30% for C. Compute the probability that two
randomly chosen very satisfied clients use the same bank.
9. There are 99 normal coins and a special one with two heads. We chose one coin, tossed it 6
times, and obtained 6 heads. What is the probability that it was one of the normal coins?
10. There are two taxi companies: Green Taxi (85 cars) and Blue Taxi (15 cars). One of the taxi
drivers caused an accident on a rainy night and escaped. The only witness testified that the
taxi was blue, but experiments showed that this particular person in these particular conditions
recognizes the colour correctly in 80% of cases, and identifies blue as green or vice versa in 20%
of cases. What is the probability that the taxi involved was really blue?
11*. (The problem may have different solutions, depending on some assumptions that are not clearly
formulated in the text).
Exactly one of the prisoners X, Y and Z will be paroled next month, but only the guards know
which one. X tried to get more information, but only one guard wanted to discuss the matter:
he said ’Y stays here’ and refused to answer any further questions. X thinks: ’previously the
probability that I’ll go out was 1/3 and now it’s 1/2 as only Z and me can be released’. X’s
cellmate says: ’but even before you talked to the guard everyone knew that Y or Z (or maybe
both) will stay in prison, and the guard only chose one of them without revealing any information
concerning you, so the probability that you’ll be freed remained unchanged; it’s 1/3 as before’.
Who is right?
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Probability and Statistics
Set 4
1. We roll a fair die 12 times. Let X denote the number of 6’s. Find the expectation of X.
2. There are eight socks in a box, of which four are white (and identical) and four are black and
identical. Socks are drawn one at a time (without replacement) until a pair is produced. What
is the expected value of the number of drawings?
3. Suppose that a player starts with 8. A fair coin is tossed three times. If the coin comes up heads,
the player’s fortune is doubled, otherwise it is halved. What is the player’s expected gain?
4. Compute the variances of the random variables defined in Ex. 1, 2 and 3.
5. Six different pairs of socks are put into a faulty washing machine. The machine damages five
socks, so that only seven remain usable. What is the expected number of complete pairs we now
have? What is the variance of the number of complete pairs remaining?(*)
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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Probability and Statistics
Set 5
1. Two points are chosen at random from a segment of length 1. Find the probability that the
distance between them is less than t.
2. Two real numbers are chosen at random from the interval [0, 3]. Find the probability that their
sum is greater than 4. Find the probability that their product is greater than 4. Find the
probability that the sum of their squares is greater than 4.
3. 51 cars going with constant velocities 50 km/h, 51 km/h, ..., 100 km/h left A for B. The fastest
car arrived in B after 2 hours. What is (approximately) the average time of the journey of a
randomly chosen car?
(Remark: It is easy to guess that it should be around 3 hours, but please try to find a better
approximation).
4. A stick was broken at a randomly chosen point. Then its longer part was again broken at a
randomly chosen point. Find the probability that the longer part of the longer part is shorter
than the originally shorter part.
(Warning: It should be stressed that the positions of the two breaking points are not independent.
Can you explain why?)(*)
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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Probability and Statistics
Set 6
In the examples below find the distibution functions (the cumulative d.f. and the density, if it exists),
the expected value and the variance of X.
1. Let P be a random point inside a circle of radius 1, and let X be the distance between P and
the centre of the circle.
2. Let P be a random point of a square of side 1, and let X be the smallest of the four distances
between P and the sides of the square.
3. Let P be a random point of a rectangle of sides 1 and 2, and let X be the largest of the four
distances between P and the sides of the rectangle.
4. A stick of length 1 was broken at a randomly chosen point. Let S and L be the lengths of the
pieces (respectively shorter and longer) and let X = S · L.
5. As above, but let X =
S
L.
6. Mr. Jones sometimes (with probability 0.4) goes to work by car, which takes him between 40
and 80 minutes (we assume that the distribution is uniform on [40, 80]). With probability 0.6,
however, Mr. Jones prefers to take a train. The train journey always lasts exactly 70 minutes.
X = Mr. Jones’s travel time.
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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Probability and Statistics
Set 7
1. The age distribution of people working in a company has the following density:
k(t − 20)(70 − t)5 , t ∈ (20, 70)
f (t) =
.
0,
t∈
/ (20, 70)
Find k. Compute the average age of an employee. Are there more employees younger or older
than average?
2. Let (X, Y ) be uniformly distributed on the triangle with vertices (0, 0), (1, 1), and (1, 2). Find
the (marginal) distributions of X and Y as well as their expectations and variances.
3. Let (X, Y ) be uniformly distributed on the square with vertices (1, 0), (0, 1), (−1, 0), and (0, −1).
Find the (marginal) distributions of X and Y . Compute the expectation and variance of Z =
2X + 3Y .
4. Random variable X is uniformly distributed on (0, 3) and Y has the two point distribution:
P (Y = 1) = P (Y = −1) = 12 . Assuming that X and Y are independent, find the distributions
Y
of the random variables X + Y , X
Y and X , and compute their expectations and variances.
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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Probability and Statistics
Set 8
1. Let X, Y , Z be i.i.d. (independent, identically distributed) random variables with density 1 on
(0, 1). Find the density of X + Y + Z.
2. Let X have the Cauchy distribution with parameter 1. Prove that Y =
bution.
1
X
has the same distri-
3. Let X and Y be i.i.d. with the normal distribution N (0; 1). Prove that for any constants a and
b the following random variables aX + bY and bX − aY are i.i.d.
4. The i.i.d. random variables X and Y have the exponential distribution with parameter a. Let
X
U = X + Y , V = X+Y
. Find the joint distribution of (U, V ). Find the marginal distribution of
V . Check the independence of U and V .
5. Let X1 , X2 , ..., Xn be n independent random variables, each of which has an exponential density
with mean m. Let X = min(X1 , X2 , ..., Xn ). Show that the density of X is exponential with
mean m
n.
Hint: Use cumulative distribution functions.
Application: A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000
hours. What is the expected time for the first of these bulbs to burn out?
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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Probability and Statistics
Set 9
1. The average number of cars stolen in a particular town is one per week. What is the probability
that at least one car will be stolen tomorrow? (here and below ignore the dependence of the
thieves’ activity on the day of week etc.)
2. The average number of cars stolen in a particular city is ten per week. What is the probability
that at least one car will be stolen tomorrow?
3. The quality of a data transmission channel is such that on average 99% of bits are received
correctly. Find the probability that there will be more than 5 errors in a message of 200 bits.
4. A Geiger-Mller counter and a source of radiation are placed in such a way that the probability
of registering a nuclear particle by the counter is equal to 0,0001. Assume that a radioactive
sample emitted 30000 particles. What is the probability of
(a) registering more than 3 particles
(b) not registering any particle
(c) registering exactly 3 particles.
5. As a manufacturer of raisin cakes we prepare 100 kg of dough, then we add some raisins, and
finally we divide the dough into 1000 cakes weighing 100 g each. Assume now that the positions
of the raisins in the dough have independent uniform distributions. As a result some cakes have
no raisin inside - we call them bad. How many raisins do we need if we require that:
(a) P (a cake is bad) < 0, 5
(b) P (a cake is bad) < 0, 01.
6. A fair coin is tossed 10 000 times. What is the probability that exactly 5000 heads are obtained?
7. An unfair coin with P (heads) = 0, 51 is tossed 10 000 times. What is the probability that exactly
5000 heads are obtained?
8. Which one is larger: the probability that exactly 50 heads are obtained in 100 tosses of a
fair coin, or the probability that exactly 51 heads are obtained in 100 tosses of a coin with
P (heads) = 0, 51?
9. Fifty small trees of a certain type were planted in sandy soil, and another fifty in clay soil. Let X
= the number of trees planted in sandy soil that survive 1 year, Y = the number of trees planted
in clay soil that survive 1 year. If the probability that a tree planted in sandy soil will survive
1 year is 0,7 and the probability of 1-year survival in clay soil is 0,6 compute an approximation
to P (|X − Y | ≤ 6).
10. In a certain group of people the height is normally distributed with mean 174 cm and standard
deviation 15 cm. We want to call the tallest 15% of the group members ’very tall’. What is the
minimal height of a very tall person?
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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Probability and Statistics
Set 10
1. Compute the correlation coefficient of the variables X and Y if their covariance is 1225, the
variance of X is 1600, and the variance of Y is 2500.
2. Let X, Y , Z be three independent random variables with variances 2, 4 and 5, respectively. Find
V (2X − 3Y + 4Z), cov(X + Y, Y + Z), cov(XY, Z).
3. Let X and Y be independent random variables with distributions N (2; 1) and N (−1; 4) respectively. Compute the correlation coefficient between X + Y and X − Y .
4. Let X and Y be independent random variables with densities exp −x and 2 exp −2y for x, y > 0.
Compute the correlation coefficient between X + Y and X − Y .
5. We roll a (usual) die 120 times. Let X be the number of 6’s, and Y - the number of odd results
(1’s, 3’s and 5’s) in 120 trials. Compute EX, EY , V X, V Y and cov(X, Y ).
6. A player throws darts at a target, scoring 0, 5 or 10 points with probability 0.1, 0.6 and 0.3,
respectively. Moreover, she wins $1 each time she scores 10 points and has to pay $2 each time
she scores 0. Let X denote the total number of points obtained by the player in 100 independent
throws, and Y - the total amount of money earned by her in the same 100 throws. Find E(X),
E(Y ), V (X), V (Y ) and cov(X, Y ).
Warsaw University of Technology, Department of Electronics and Information Technology
ECE, 2008/2009, winter semester
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