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Bài tập toán cao cấp chapter 2 exercises

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CHAPTER 2 EXERCISES
1. An unbiased disc has a single dot marked on one side and two dots marked
on the other side. The disc and an unbiased die are thrown and the random
variable X is the sum of the numbers of dots showing on the disc and on the
top of the die. Tabulate the probability distribution of X.
Show that P (X ≥ 4|X

7) = 8/11.

Write down E(X) and show that Var(X)=19/6.
Two independent observations X1 and X2 are taken of X.
a. Find V ar(X1 − X2 ).
b. Find P (X1 − X2 ≥ 5).
2. A computer generates a random variable X whose probability distribution
is given in the following table
x
0
2
4
6

P(X=x) 0.1 0.2 0.3 0.4
Show that V ar(X) = 4.
Find E(X 4 ) and V ar(X 2 ).
Two independent observations of X are denoted by X1 and X2 . Show that
P (X1 + X2 = 6) = 0.2 and tabulate the probability distribution of X1 + X2 .
The sum of 100 independent observations of X is denoted by S. Describe
fully the approximate distribution of S.
3. An unbiased cubical die has three faces numbered ’1’, two faces numbered
’2’ and one face numbered ’3’. The random variable X is the number showing
on the top face of the die when it is thrown. Show that E(X) = 5/3, and
find V ar(X).
4. A computer can give independent observations of a random variable X with
probability distribution given by P (X = 0) = 3/4 and P (X = 2) = 1/4.
It is programmed to output a value for the random variable Y defined by
Y = X1 + X2 , where X1 and X2 are independent observations of X. Tabulate
the probability distribution of Y, and show that E(Y ) = 1.
The random variable T is defined by T = Y 2 . Find E(T ) and show that
V ar(T ) = 63/4.


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The computer is programmed to produce a large number n of independent
values for T and to calculate the mean M of these values. Find the smallest
value of n such that P (M < 3) > 0.99.
5. The discrete random variable X has probability distribution as shown in
the table below, where p is a constant.
x
0 1
2
3
P(X=x) p p/2 p/4 p/20
a. Show that p = 5/9.
b. Find E(X), Var(X). Take a random sample of X with sample size 100 and
denote by Y the number of observations on which X = 3. Using a suitable
approximation, show that P (Y = 2) = 0.240 and P (Y ≥ 4) = 0.303
(correct to 3 places of decimals).
6. In a game, 2 red balls and 8 blue balls are placed in a bottle. The bottle
is shaken and Mary draws 3 balls at random and without replacement. The
number of red balls that she draws is denoted by R. Find the probability
distribution of R, and show that P (R ≥ 1) = 8/15.

Show that the expectation of R is 3/5 and find the variance of R.
Mary scores 4 points for each red ball that she draws. The balls are now
replaced in the bottle and the bottle is shaken again. John draws 3 balls at
random and without replacement. He scores 1 point for each blue ball that
he draws. Mary’s score is denoted by M and John’s score is denoted by J.
Find the expectation and variance of M-J.
7. Alfred and Bertie play a game, each starting with each cash amounting to
100 pound. Two dice are thrown if the total score is 5 or more then Alfred
pays x pound, where 0 < x

8, to Bertie. If the total score is 4 or less then

Bertie pays x+8 pound to Alfred. Show that the expectation of Alfred’s cash
after the first game is (304-2x) pound.
Find the expectation of Alfred’s cash after six games.
Find the value x for the game to be fair, i.e., for the expectation of Alfred’s
winnings to equal the expectation of Bertie’s winnings.
Given that x=3, find the variance of Alfred’s cash after the first game.


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8. Emergency calls to an ambulance service are received at random times, at
an average rate of 2 per hour. Calculate the probability that
a. more than 3 emergency calls are received in a randomly chosen one-hour
period,
b. exactly 3 emergency calls are received in a randomly chosen two-hour
period.
Twelve one -hour periods are chosen at random, and the number of emergency
calls received in each of the periods is recorded. Find the probability that
more than 3 calls are recorded in at least two of the twelve periods.
9. A random variable X has the probability distribution given in the following
table
x

2 3

4

5

P(X=x) p 2/10 3/10 q
a. Given that E(X) = 4, find p and q.
b. Show that Var(X)=1.
c. Find E(|X − 4|).
d. Ten independent observations of X are taken. Find the probability that
the value 3 is obtained at most three times.
10.
a. The random variable X has a Poisson distribution with mean a. Given
that
P (X = 1) = 3P (X = 0),
find the value of a, and hence calculate P (X > 2), giving 3 decimal places
in your answer.
b. The random variable S is the number of successes in 5 independent trials
in which the probability of success in any trial is 1/3, so that S has a
binomial distribution with n=5 and p = 1/3. The random variable D is
the difference (taken always as possible) between the number of successes


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and the number of failures in 5 such trials; hence D can take the values
1, 3, 5 only.
i. Show that P (D = 1) = 40/81, and find P (D = 3) and P (D = 5).
ii. Find E(D2 ).
It can be shown that D2 = 4S 2 −20S+25. Use standard results concerning
the mean and variance of a binomial distribution to obtain the values of
E(S) and E(S 2 ), and hence check the value of E(D2 ) found in part ii.
11. A writer who writes articles for a magazine finds that his proposed articles
sometimes need to be revised before they are accepted for publication. The
writer finds that the number of days, X, spent in revising a randomly chosen
article can be modelled by the following discrete probability distribution.
Number of days, x 0
1
2
4
P(X=x)
0.8 0.1 0.05 0.05
Calculate E(X) and Var(X).
The writer prepares a series of 15 articles for the magazine. Find the
expected value of the total time required for revision to these article.
The writer regards articles that need no revisions (i.e. for which X=0)
or which need only minor revisions (X=1) as ’successful’ articles, and those
requiring major revisions (X=2) or complete replacement (X=4) as ’failures’.
Assuming independence, find the probability that there will be fewer than 3
’failures’ in the 15 articles in the series.
The writer produces 50 articles. Use an approximate Poisson distributed
to find the probability that at least 2 of these 50 articles will need to be
completely replaced.
12. A circular card is divided into 3 sectors scoring 1, 2, 3 and having angles
135o , 90o , 135o , respectively. On a second circular card, sectors scoring 1, 2, 3
have angles 180o , 90o , 90o respectively. Each card has a pointer pivoted at its
center. After being set in motion, the pointers come to rest independently in
random positions. Find the probability that
a. the score on each card is 1,
b. the score on at least one of the cards is 3.


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The random variable X is the larger of the two scores if they are different,
and their common value if they are the same. Show that P (X = 2) = 9/32.
Show that E(X) = 75/32 and find V ar(X).
13. Serious delay on a certain railway line occur at random, at an average
rate of one per week. Show that the probability of at least 4 serious delays
occurring during a particular 4-week period is 0.567, correct to 3 decimal
places.
Taking a year consists of thirteen 4-week periods during which at least 4
serious delays occur.
Given that the probability of at least one serious delay occurring in a
period of n weeks is greater than 0.995, find the least possible integer value
of n.
14. The probability that a randomly chosen flight from Stanston Airport is
delayed by more than x hours is

1
100 (x

− 10)2 , for x ∈ R, 0

x

10. No

flights leave early, and none is delayed for more than 10 hours. The delay, in
hours, for a randomly chosen flight is denoted by X.
a. Find the median, m, of X, correct to three significant figures.
b. Find the cumulative distribution function, F , of X and sketch the graph
of F .
c. Find the probability density function, f , of X and sketch the graph of f .
d. Find E(X).
A random sample of 2 flights is taken. Find the probability that both flights
are delayed by more than m hours, where m is the median of X.
15. The continuous random variable X has a uniform (rectangular) distribution on 0

x

a. Find the cumulative distribution function of X.

Two independent observations, X1 and X2 , are made of X and the larger
of the two values is denoted by L.
i. Use the fact that L < x if and only if both X1 < x and X2 < x to find the
cumulative distribution function of L.


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ii. Hence show that the probability density function of L is given by

 2x2 0 x a,
f (x) = a
0 otherwise.
iii. Hence find E(L) and Var(L).
iv. Find also the median of L.
16. The random variable X has a normal distribution and
P (X > 7.460) = 0.01, P (X < −3.120) = 0.25.
Find the standard deviation of X. 200 independent observations of X are
taken.
a. Using a Poisson approximation, find the probability that at least 197 of
these observations are less than 7.460.
b. Using a suitable approximation, find the probability that at least 40 of
these observations are less than -3.120.
17. The continuous random variable X has probability density function f
given by

k(2 − x),
f (x) =
0,

for 0

x

2,

otherwise,

where k is a constant.
a. Find the value of k.
b. Find the cumulative distribution function of X.
The continuous random variable Y is given by Y = 1 − 12 X. Show that
P (Y < y) = y 2 , where 0

y

1.

Deduce the probability density function of Y and hence, or otherwise,
show that E(Y ) = 2/5.


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18. The continuous random variable X has probability density function given
by
f (x) =


k

x

1

0

x

e,

otherwise,

where k is a constant.
a. Show that k=1.
b. Find E(X) in terms of e, and show that V ar(X) = 12 (3 − e)(e − 1).
c. Find the cumulative distribution function F of X, and sketch the graph of
y = F (x).
d. Find E

1
X

in terms of e.

19. The independent random variables R and S each have normal distributions. The means of R and S are 10 and 12 respectively, and the variance are
9 and 16 respectively. Find the following probabilites, giving your answers
correct to 3 significant figures.
a. P (R < 12).
¯ < 12), where R
¯ is the mean of a sample of 4 independent observations
b. P (R
of R.
c. P (R < S).
d. P (2R > S1 + S2 ), where S1 and S2 are two independent observations of S.
20. The continuous random variable X has probability density function given
by
f (x) =




k
(x+1)4 ,

0,

for x ≥ 0,
for x < 0,

where k is a constant.
a. Show that k = 3, and find the cumulative distribution function. Find also
the value of x such that P (X

x) = 7/8.


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b. Find E(X + 1), and deduce that E(X) = 1/2.
c. By considering V ar(X + 1), or otherwise, find V ar(X).
21. The mass of coffee in a randomly chosen jar sold by a certain company
may be taken to have a normal distribution with mean 203 g and standard
deviation 2.5 g.
a. Find the probability that a randomly chosen jar will contain at least 200
g of coffee.
b. Find the mass m such that only 3% of jars contain more than m grams
of coffee.
c. Find the probability that two randomly chosen jars will together contain
between 400 g and 405 g of coffee.
d. The random variable Y denotes the mean mass (in grams) of coffee per
jar in a random sample of 20 jars. Find the value of a such that
P (|Y − 203| < a) = 0.95.
22. The continuous random variable X is such that



1
x 0,


P (X > x) = k(3 − x)3 0 x



0
x ≥ 3.

3,

Show that P (X > 1) = 8/27. Find the probability density function of X, and
hence find E(X). 108 independent observations are taken of X.
i. The number of observations greater than 2 is denoted by N. Using a suitable
approximation, find P (3 < N < 6).
ii. The number of observations greater than 1 is denoted by M. Using a
suitable approximationl find P (24 < M < 40).
23. The random variable X has a normal distribution with mean 3 and variance 4. The random variable S is the sum of 100 independent observations
of X, and the random variable T is the sum of a futher 300 independent
observations of X. Giving your answers to 3 places of decimals, find


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a. P (S > 310).
b. P (3S > 50 + T ).
The random variable N is the sum of n indenpendent observations of X. State
the approximate value of P (N > 3.5n) as n becomes very large, justifying
your answer.
24. The continuous random variable X has a uniform (rectangular) distribution on the interval [10,20], i.e. the probability density function f is given
by
f (x) =


1

for 10

0

otherwise .

10

x

20,

Write down the value of E(X), and show by integration that V ar(X) = 25/3.
The cumulative distribution function of X is F. Express F(x) in terms of
x, for 10

x

20. Sketch the form of the graph of F for all values of x.
1
1
The random variable Y is defined by Y = 2 . Show that E(Y ) =
,
X
200
and find Var(Y).
25. The random variable X has probability density function given by:



kx, 0 x 1,


f (x) = k,
1 < x 2,



0,
otherwise

where k is a constant.
a. Show that k = 2/3.
b. Find E(X) and E(X 2 ).
c. Show that the median m of X is 1.25, and find P (|X − m| > 1/2).
26. The random variable T has probability density function given by

k(t − 2) 2 t 4,
f (t) =
0
otherwise,
where k is a constant. Find, in terms of k, the cumulative distribution function. Hence, or otherwise, find the value of k.


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Show that P (T > 3) = 3/4.
Find E(T ) and V ar(T ).
The event T > 3 is denoted by A and the event 2T > 3 is denoted by B.
Find P (A ∪ B) and P (A ∩ B).
27. The continuous random variable X has probability density function given
by
f (x) =




1/2 −1



x

0,

1/4 0 < x 2,



0
otherwise.

A sketch of the graph of the probability density function is given above. Show
that

1 1
+ x, for 0 < x
2 4
and find a similar expression for P (X x), for −1
P (X

x) =

2,
x

0.

Show that E(X) = 1/4, and state the value of E(X − 14 ).

Find E( X + 1).
10 independent observations of X are taken. Find the probability that 8
of these observations are less than 1.
28. The random variable X has probability density function f given by

 3 (4 − x2 ) −2 x 2,
f (x) = 32
0
otherwise,
Show that V ar(X) = 4/5.
The random variable Y is defined by Y = aX + b, where a and b are
positive constants. It is given that E(Y)=50 and Var(Y)=80. Find a and b.
A random sample constists of 160 independent observations of Y. Find
an approximate value for the probability that the sample mean lies between
49.0 and 50.5.
29. The continuous random variable U has a uniform distribution on 0 <
u < 1. The random variable X is defined as follows
X = 2U when U

3
3
, X = 4U when U > .
4
4


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a. Give a reason why X cannot take values between
the values of P (0 < X

3
2

and 3, and write down

3/2) and P (3 < X < 4).

b. Sketch the complete graph of the probability density function of X.
c. Find the lower quartile q of X, i.e. the value of q such that P (X < q) = 1/4.
d. Three independent observations are taken of X. Find the probability that
they all exceed q.
e. Show that EX =

23
and find E(X 2 ).
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30. The continuous random variable X has probability density function f
given by

k( 2 − 1 ),
x
2
f (x) =
0,

for 1

x

2,

otherwise ,

where k is a constant.
a. Show that k=2.
b. Find the cumulative distribution function, F, of X, and hence ore otherwise
find the value of t for which F (t) = 2/3.
c. Find the mean of X and show that the variance of X is approximately
0.0472.
31. The continuous random variable X has cumulative distribution function
F given by



0
x < 0,


F (x) = a(6x2 − x3 ) 0 x



1
x > 4,

4

where a is a constant. Find a.
Find the probability density function of X, and use the fact that the graph
of this function is symmetrical about x=2 to write down the expectation of
X.
Show that the variance of X is 4/5.


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The random variable S is the sum of 100 independent observations of X.
Find P (X > 185).
32. The continuous random variable X is the distance, measured in hundreds
of kilometres, that a particular car will travel on a full tank of petrol. It is
given that



0
x


x) = ax2 − 8ax + b 3



1
4

P (X

3,
x

4,

x,

where a and b are constants. Show that a=-1, find the value of b, and verify
that P (X

3.5) = 3/4.

a. Find the probability density function of X.
b. Show that E(X) =

10
.
3

c. Three independent observations of X are taken. Find the probability that
two of the observations are less than 3.5 and one is greater than 3.5.
d. One hundred independent observations of X are taken and M is the arithmetic mean of the observations. Given that V ar(X) = 1/18, state the
approximate distribution of M.
33. Small packets of nails are advenised as having average weight 500 g, and
large packets as having average weight 1000 g. Assume that the packet weights
are normally distributed with means a advenised, and standard deviation of
10 g for a small packet and 15 g for a large packet. Giving your answers
correct to 3 decimal places.
a. find the probability that a randomly chosen small packet ha a weight
between 495 and 510 g.
b. find the probability that two randomly chosen small packets have a total
weight between 990 g and 1020 g.
c. find the probability that the weight of one randomly chosen large packet
exceeds the total weight of two randomly chosen small packets by at least
25 g.


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d. find the probability that one half of the weight of one randomly chosen
large packet exceeds the weight of one randomly chosen small packet by
at least 12.5 g.
34. The continuous random variable X has cumulative distribution function
F given by



0



F (x) =
x



1

for x

0,

for 0 < x < 1,
for x ≥ 1.

Find the median of X.
The probability density function of X is f. Write down an expression for
f(x), valid for 0 < x < 1. Hence
a. show that E(X) = 1/3,
b. find Var(X).
Show that the median of



X and mean of



X are equal.

35. The continuous random variable X has probability density function given
by

ax + b 0 x 2
f (x) =
0
otherwise
where a and b are positive constants. It is given that P (X

1) = 1/3. Find

the values of a and b.
Show that E(X) = 16/9 adn find E(X 2 ).
Six independent observations of X are taken and the random variable R
is the number of observations such that X

1. Find E(R) and P (R

2).

36. A machine makes metal rods. A rod is oversize if its diameter exceeds
1.05 cm. It is found from experience that 1% of the rods produced by the
machine are oversize. The diameters of the rods are normally distributed with
mean 1.00 cm and standard deviation σ cm. Find the value of σ, giving 3
decimal places in your answer.
Two hundred rods are chosen at random. Using a suitable approximation,
find the probability that four or more of the rods are oversize, giving 3 decimal
places in your answer.


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Another machine makes metal rings. The internal diameters of the rings
are normally distributed with mean (1.00+2σ) cm and standard deviation 2σ
cm, where σ has the value found in the first paragraph. Find the probability
that a randomly chosen ring can be threaded on a randomly chosen rod,
giving 3 decimals places in your answer.
37. A householder wishes to sow part of her garden with grass seed. She
scratters seed randomly so that the number of seeds falling on any particular
region is a random variable having a Poisson distribution, with its mean
proportional to the area of the region. The part of the garden that she intends
to sow has area 50m2 and she estimate that she will sow 106 seeds. Calculate
the expected number of seeds falling on a region , R, of area 1cm2 and show
that the probability that no seeds fall on R is 0.135, correct to three significant
figures.
The number of seeds falling on R is denoted by X. Find the probability
that either X=0 or X ≥ 4.
The number of seeds falling on a region of area 100cm2 is denoted by Y.
Using a normal approximation, find P (175

Y

225).

38. A fair coin is tossed 10 times. Calculate the probability that at least 3
heads are obtained.
The coin is tossed 80 times. Use the suitable approximation to find the
probability that the number of heads obtained lies between 35 and 45 inclusive.
39. Two red balls and two white balls are placed in a bag. Balls are drawn
one by one, at random and without replacement. The random variable X is
the number of white balls drawn before the first red ball is drawn.
a. Show that P (X = 1) = 1/3, and find the rest of the probability distribution of X
b. Find E(X) and show that V ar(X) = 5/9.
¯
c. The sample mean for 80 independent observations of X is denoted by X.
¯ > 0.75).
Using a suitable approximation, find P (X
40.


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a. Plane crashes involving loss of life may be assumed to occur at random
at an average rate, for all major airlines taken together, of 1.5 per year.
Find, correct to 3 decimal places, the probability of there being
i. at most 2 such crashes in a one - year period.
ii. exactly 4 such crashes in a two - year period.
b. When a machine is used to dig up potatoes there is a probability 0.1 for
each individual potato that it will be damaged in the process.
i. Find, correct to 3 decimal places, the probability that a random selection of 12 potatoes dug up by machine will include at least 3 damaged
ones.
ii. A random sample of n potatoes is selected, and the number of damaged
potatoes in the sample is denoted by the random variable X. Write
down expressions, in terms of n, for the mean and standard deviation
of X.
Use a normal approximation to show that, for P (X ≥ 3) to exceed
0.95, the sample size n must satisfy the (approximate) inequality.

n − 25 > 4.935 n.
41. A shop receives a batch of 1000 cheap lamps. The odds that a lamp is
deffective is 0.1%. Let X be the number of deffective lamps in the batch.
a. What kind of distribution does X have? What is/are the value(s) of parameter(s) of this distribution?
b. What is the probability that the batch contains no deffective lamps? One
deffective lamp? More than two deffective ones?
42. Early in the morning, a group of m people decides to use the elevator
in an otherwise deserted building of 21 floors. Each of these persons chooses
his or her floor independently of the others, and - from our point of view completely at random, so that each person selects a floor with probability


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1/21. Let Sm be the number of times the elevator stops. In order to study
Sm , we introduce for i = 1, 2, ..., 21 random variables Ri , given by

1 if the elevator stops at the ith floor,
Ri =
 if the elevator does not stop at the ith floor.
a. Each Ri has a Bernoulli distribution with parameter p. Show that p =
1 − (20/1)m .
b. From the way we defined Sm , it follows that
Sm = R1 + · · · + R21 .
Can we conclude that Sm has a B(21, p) distribution, with p as in part
a? Why or why not?
c. Clearly, if m = 1, one has that P (S1 = 1) = 1. Show that for m = 2
P (S2 = 1) = 1/21 = 1 − P (S2 = 2)
and that S3 has the following distribution
a

1

2

3

P (S3 = a) 1/441 60/441 380/441
43. You decided to play monthly in two different lotteries, and you stop
playing as soon as you win a prize in one (or both) lotteries of at least one
million euros. Suppose that every time you participate in these lotteries, the
probability to win one million (or more) euros is p1 for one of the lotteries
and p2 for the other. Let M the number of times you participate in these
lotteries until winning at least one prize. What kind of distribution does M
have and what is its parameter?



x

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