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.. The discrete random variable X has probability dis
Trang 1CHAPTER 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 6 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
P(X=x) 0.1 0.2 0.3 0.4
Show that V ar(X) = 4
Find E(X4) and V ar(X2)
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 = Y2 Find E(T ) and show that
V ar(T ) = 63/4
Trang 2The 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
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 6 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
Trang 38 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
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
Trang 4and 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 = 4S2−20S+25 Use standard results concerning the mean and variance of a binomial distribution to obtain the values of E(S) and E(S2), 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
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
Trang 5The 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 1001 (x − 10)2, for x ∈ R, 0 6 x 6 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) distribu-tion on 0 6 x 6 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
Trang 6ii Hence show that the probability density function of L is given by
f (x) =
2x
a 2 0 6 x 6 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
f (x) =
k(2 − x), for 06 x 6 2,
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 − 12X Show that
P (Y < y) = y2, where 0 6 y 6 1
Deduce the probability density function of Y and hence, or otherwise, show that E(Y ) = 2/5
Trang 718 The continuous random variable X has probability density function given by
f (x) =
k
x 1 6 x 6 e,
0 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 X1 in terms of e
19 The independent random variables R and S each have normal distribu-tions 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)
b P ( ¯R < 12), where ¯R is the mean of a sample of 4 independent observations
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, for x ≥ 0,
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 6 x) = 7/8
Trang 8b 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
P (X > x) =
k(3 − x)3 0 6 x 6 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 vari-ance 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
Trang 9a 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) distri-bution on the interval [10,20], i.e the probability density function f is given by
f (x) =
1
10 for 10 6 x 6 20,
0 otherwise 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 6 x 6 20 Sketch the form of the graph of F for all values of x The random variable Y is defined by Y = 1
X2 Show that E(Y ) = 1
200, and find Var(Y)
25 The random variable X has probability density function given by:
f (x) =
kx, 0 6 x 6 1,
k, 1 < x 6 2,
0, otherwise where k is a constant
a Show that k = 2/3
b Find E(X) and E(X2)
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
f (t) =
k(t − 2) 2 6 t 6 4,
where k is a constant Find, in terms of k, the cumulative distribution func-tion Hence, or otherwise, find the value of k
Trang 10Show 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 6 x 6 0, 1/4 0 < x 6 2,
0 otherwise
A sketch of the graph of the probability density function is given above Show that
P (X 6 x) = 1
2 +
1
4x, for 0 < x 6 2, and find a similar expression for P (X 6 x), for −1 6 x 6 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
f (x) =
3
32(4 − x2) −2 6 x 6 2,
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 6 3
4, X = 4U when U >
3
4.
Trang 11a Give a reason why X cannot take values between 32 and 3, and write down the values of P (0 < X 6 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
16 and find E(X
2)
30 The continuous random variable X has probability density function f given by
f (x) =
k(x2 − 12), for 1 6 x 6 2,
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
F (x) =
a(6x2 − x3) 0 6 x 6 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