The American Term Life Insurance History Project

You are reading a page from An Elementary Treatise on Actuarial Mathematics by Harry Freeman (1932)
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Term Life Insurance

MISCELLANEOUS EXAMPLES
1. Find the sum of n terms of the series 1, 2, 4, 9, ¹9, 36, 62, ....

2. A person writes four letters and four envelopes. If the letters are placed in the envelopes at random, what is the chance that not more than one letter is placed in its correct envelope?
3. Make a rough sketch of the curve y² = x² (1 — x²). Find the maximum and minimum values of y and the area enclosed by the curve.

4. Given It_o = 1027, U₆ = 1212, U₁₂= 1469, u₁y = 2014, explain

(i)how you would complete the series from uo to u₆i

(ii) how you would proceed if you were asked to complete the series from u

₆

to u

₁₂

supposing that it were unnecessary to

find

₁

₂

, u

₃

₄

and u

₅

Differentiate

(i) (x

x) ;

(ii) sin-

(sin-

x).

6. Evaluate(i) Lt

^x4

—

2x3

— 2X

—

cose

—

cot

(u)

^Lt

e=,o6

7. The faces of a cubical die are marked

2, 2, 4, 6, 6. Find the chance that in ten throws, four 2's, two 4

four 6's are thrown.

8. Integrate

2x4

3 dx

and

dx.

9. Find the tenth term of the series:

^(a)¹, 4, 13, 36, 97, ^268,765, ... ;

2, 12, 36, 98, 270, 768, ... . lo. Show that

,\u

_,x

^Ou

+ ux+l

and hence prove by

matheinduction

that

uxvx

ⁿ

+ n,vrO

^''-l

ux+

ⁿ

^,2v

^An-2u

s+2

₃

vxA

^n-3

u,.+

₃

+ ....

II. A policy register of 384 pages contains particulars of 1920 policies, an equal number being entered on each page. 640 of the whole number are policies for 5oo, and 480 of the whole number have terOn how many pages of the register would you expect to find

(a)

at least one policy for

500

in force;

500

in force;more than one policy for

500

in force?

24-2

372    ACTUARIAL MATHEMATICS
12. Differentiate
    tan-' bx — a sin-I    3 + ⁴'    cos -1 I — x²
ax + b'    _S'\/I .+ _. x2'    1 + x²

13. Prove that in the process of obtaining divided differences of the function u_s , given u_a , u,, , u_c, ... , the last divided difference is numerically the same whatever the order of the arguments and the corresponding u's.

14. Given that
(^I) ^u-1⁼4;^u1⁼⁶;
(2) the area between the curve y = u_s, the x-axis and the ordinates u_₁ and uo is 4.7 ;
(3) the tangent to the curve y = u, at the point (o, u_s) makes an angle 0 with the x-axis such that tan 6 = •8 ; find an approximate value for u_s.
15. Show that
x(2) x(3)
(I) Exm=C+ ₂₎.~O^m+-~ A²o^m+...; 3•
x(2) x(3)
(2) Eu_x = C + x^u)u_o + z Dun + ⁱ3•
A²u₀ + ... .

16. Two Companies A and B make simultaneous issues each of moo bonds. Those of Company A are redeemable by equal drawings spread over 20 years, and those of B by equal drawings spread over 40 years. Find, in the case of two definite bonds, one of each issue:

(I) the probability that the bond of Company B is redeemed before the bond of Company A ;

(2) the probability that the bond of Company B is redeemed before
the bond of Company A and within 15 years of issue.

17. Prove that dx -- = I
J_ log (2 + 1/3).
JO I+zcosx ₁.₃

18. If us, u_s, u₁₀, u₁, be four values of a function at equidistant points, find expressions true to third differences for u₆ and us, solely in terms of u_s, us, _uloand u,₅.
19. The area of a curve is given by A = y 1/(z5 + 4y) (4 — y). Plot A against y on squared paper and hence obtain the maximum value of A and the value of y for which A is a maximum. Verify your results by the methods of the calculus.

MISCELLANEOUS EXAMPLES 373

Define the following types of functions, giving examples: Inverse function; Rational Integral function; Multiple-valued function; Algefunction.

—

Prove that

^{q (x)}

^(v)

( )- -

^-x'

)(y)

Two throws are made, the first with three dice and the second with two. What is the probability both that the first throw is not less than II and that the second throw is not less than 8?

Prove that log

(1 —

^-4

is positive for all values of x between o and unity.

A horizontal trough with vertical ends is of V-shaped cross-section, the angle between the sides being 6o°, and the length of the trough 6 feet. If water enters at the rate of 4 cu. ft. per min., find the rate at which the surface is rising when the depth is

foot.

Show that the series whose nth term is

( I)^n-1I .3.5_... (2n - 3) A2(n-1) ^ux-n+} 8n-1 (n - I)1
is equivalent to

2 (u_x - ¹¹x+1 + ¹¹x+2 - 11x--₃ + ...).

25.Obtain the approximate quadrature formula

r14

I_}nxdz

(2711

₀

1711

₁

511

₂

—

₃

26.The numbers of members in a Friendly Society were available

for the following years:Year	1922	1923	1924	1925	1928
Number of members	995	99⁸	1003	99⁶	97⁶

It was desired to obtain estimates for the years 1926 and 1927. This was effected on the assumption of a constant fourth difference. Subit was discovered that the numbers for 1926 were actually 1002, and a fresh estimate for the year 1927 had to be prepared. Calcuthe original estimates for 1926 and 1927, and find the revised figure for the year 1927.

27.If b and c are positive quantities (b > c) and if^3x2—bemay

6x—b3c

have any value between b and c, all such values being equally likely, find the probability that x is real.

28.If m and n are positive integers, find by successive integration

by parts the value of

(I —

ⁿ

)

dx.

374    ACTUARIAL MATHEMATICS
By expanding the integrand and integrating each term, deduce the value of the sum of the series
I    111₁    nt.,    m₃    m,,
n n+I⁺n+2 n+3 +...+(— 1)'ⁿ
n+m^'
where    m!/r! (m — r)!.

29.If log 0 = n log t — x' 4^t

~, find what value of

will make

0 2

0 20

.x-a+4

30.If E u , = w_a for all integral values of a, prove that, to the third

⁼

order of differences,

₇

= •2w; —

•

oo8 (w,o —

₅

wo).

Given the following table, find u, , u

₁₂

and u

₁₇

.r=a+4

x=a

⁰

7 '

⁶

459 •3408

'43

31.Transform the integral I ^ax² (a² — x²)^Idx by the substitution

cos

and find its value, explaining by reference to a diagram what are the new limits of integration.

A reservoir has plane sloping sides and ends; its top and base are horizontal rectangles of sides 24 ft., 16 ft. and 12 ft., 8 ft. respecand its depth is 40 ft. If water flows into it at the uniform rate of 30 cu. ft. per minute, at what rate is the surface rising when the depth of the water is lo ft.?

33•

The lengths of day on March

19,

April

18,

May

and June

are

hours, 14 hours,

hours

₄₀

minutes and

hours

₃₀

minutes respectively. Obtain an equation in the form

f (x),

where

is the length of day and x is the number of days elapsed since March

19,

and apply it to ascertain the mean length of day from March

to June

17,

both days inclusive.

34.

If events

and C are independent of each other, and events E and

are mutually exclusive and are both contingent upon the happening of A, give an expression for the probability that either E or

will

happen and that neither

nor C will happen.

MISCELLANEOUS EXAMPLES 375

The following formulae for approximate integration are correct to third differences:

•3

dx =

(

^u_

+ 2U

₀

_U2

uxdY

(tl_

₃

4uo

₃

Prove that if these formulae are applied to a function whose fifth differences are constant, the respective errors involved in the approximaare in the ratio 7 : 18, and are in opposite directions.

By a combination of the two formulae obtain an expression, correct

-3

to fifth differences, for

u,dx.

Draw the graph of y = x³ — 3X + I. By reference to the graph, supplemented by arithmetical trials, find approximately the value of the negative root of the equation x³ — 3X + I = o, correct to two places of decimals.

Define a differential coefficient. If Ax is a finite increment, is it ever true that ^Ayis equal to d_x?

If V be the volume of a regular polyhedron, and x the length of an

edge, what is the meaning of

dx'

and of

d ?

Illustrate with a regular

tetrahedron.

^—

₁

38.

₁

112 = 44;= 25 ; ~

dx = —

11.

Find u

₀

A and B throw in turn with two dice, A having the first throw. A is to win either (1) if he throws a double six in his first six throws or (2) if he throws a double six before B has a throw scoring 9 or more. Find an expression for A^'s chance of winning.

Find the mean length of a straight line drawn from one of the angular points of an equilateral triangle to a point taken at random in any one of the sides.

41.Find an expression, correct to fourth differences, tor the value

_dx

when

1, in terms of u_

_2i

, uo, u

₁

, u

₂

42. Evaluate (¹x² tan-¹ xdx and _I2sin⁴ xdx. o .o

376 ACTUARIAL MATHEMATICS

Three numbers are selected at random, one at a time, from the five numbers I, 2, 3, 4 and 5, repetitions being allowed. Find the probthat the third number selected is not less than the second and the second is not less than the first.

By means of the formula (taking n = lo)

x=mn

s=m

n —

a+x

= n

a+nx —

a+mn +

¹¹

x=0x=0

(

- - r du

_x\

J)approx.,

_x=atmn

xaa

find the approximate value of log10

9 !,

given that lo

₁₀

2 = •3010,

^lo

glo 3 = '477

^lo

glo

'4343

If x = tan 0, — 4, and y = sec , prove that (dxl/ + y³(_dx) o.

Prove that r- dx= 2 log -_ + K.

- x'fix+a Va

+a+Va

Hence

prove that

+ x2))

x \

+ K.

J(I —

)

+ x

⁴

2 logI — x

Show that x² — 2x + 4 log (x + 2) increases with x from x = — 2 to x = — 1, then diminishes from x = — 1 to x = o and then increases.

Given that u

= 16;

₁

₂

=64; u

₃

₄

₅

=266; u

+u,+u

₃

+11

₉

=1029,

find the values of u

₄

and u

₅

on the assumption that

is constant.

Through two points taken at random in a diagonal of a square, two straight lines are drawn parallel to one of the sides and to each other. Find the probability that the area of that part of the square between the two lines is not less than one-third of the area of the whole square.

50. If

be a positive integer find the limit when

-> oo of

[(I

+ n~

+ n)

+ n) ...

+ 11)J

^IIn.

Two straight roads meet at X at an angle of sin-¹ . A is travelling in the direction of X along one of the roads at 40 miles an hour, and B is walking along the other road, also towards X, at the rate of 4 miles an hour. When A is 62 miles from X, B is 81 miles from X. Find their minimum distance apart.

MISCELLANEOUS EXAMPLES 377

A and B play a match of seven games. A's chances of winning, drawing and losing any game are as 5 : 3 : 2. One point is scored for a win and half a point for a draw. Find the chance that the match is drawn.

53.Prove that u₁ — u₂ + u₃ — ... = I^li, — (2)^2Au1 + (2)3A2n1 — ...,

where

is a real positive quantity which diminishes as

increases, and Lt

n-+

In the series

1 - - + — + ...

find the value of

and prove that this series is equivalent to the series

iI+1+1.2+1.2.3+...

233

54.
Use the conception of finite differences to prove that the general term in the recurring series g_o + u₁x + u₂x² + u₃x³ + ... (scale of relation i — px — qx²) is of the form Aaⁿ + Bbⁿ, where a and b are functions of p and q, and A and B are constants.

Prove that every series whose coefficients form an arithmetical pro

gression is a recurring series, and that the generating function is

—

a) x

(1 — x)

where

is the first term and

the common difference of the pro

55.Given u_₂ = 4, u₀ = 6^.5, u₂ = 6'3, and that u_x has a maximum value when x = 1, find an approximate value for u₁.

A bag contains four black balls and eight white balls. Two balls are drawn at a time and replaced, this operation being performed six times. Calculate the probability that two black balls are not drawn four times consecutively.

57.

Find the value of

where

has the values

(i)

1;(ii)

;

(iii) e

^2x

;

(iv)

58.If x = y³ + 3a²y for real values of y, find by means of Maclaurin's theorem the expansion of y in powers of x as far as the term involving x³.
59.
Obtain a formula for the finite integration of any rational integral function of x and apply it to find the sum to n terms of the series whose rth term is (r² + 1) (r — 2).

6o.

25:3o

311;

3o:3c =

5'7

⁸

2s:35 =

66o;

35:30

14'420;

14'8

:35

15'209. Find as accurately as possible a

_27:32

378 ACTUARIAL MATHEMATICS

61. A and B toss a coin in turn, a head counting two and a tail one. The winner is the person who first scores a total of exactly three. If either tosses a head when his score is already two, his score is reduced to one. Calculate A's chance of winning the game if he has the first toss.

62. Two points are chosen at random on the circumference of a circle of radius r. Find the chance that the length of the chord joining them
is less than r 1/3.
log t
63. Find the value of d / e^txdx, where t is independent of x. dt.o
64. If A = x/(2x + z) and B = — _-1²y/x, prove that

a2A a2B
az² = az ay '

65. A person X walks along the diagonal of a square field ABCD from B to D at the uniform rate of 5 feet per second. A second person Y proceeds along the side of the field from B to C at such a rate that the positions of X and Y at any moment lie on a straight line which, when produced, would pass through A. At what speed is Y moving when X has walked one-fourth of the distance from B to D?
66. A die whose sides are marked 1, 2, 3, 4, 5, 6 is thrown five times. Find the probabilities:

(a)that the product of the five throws is 432;

(b) that the sum of the first three throws is exactly three more than the sum of the last two throws.

67. Show that the infinite series

1 — ;

+ —+ 3

s —+ ... can

be expressed in the form

1—x

⁴

1—x

and hence deduce its value.

68.Complete the series u_; to tt_isby means of Everett's formula: x— 505to152025

61•o91'4113'6

34'

79'4. 238'0296

69.Ifu₀ + u₁x + u₂x² + u₃x³ + ... = f (x),

show that

vx + ul vx+i + u2vx+2 + ... = f (1) vx + f ' (I) Av +f"

^(I)

The probability that A will die within ten years is •2 and the probability that A, B, C will all be alive ten years hence is •42. The probability that at least one of the three will be alive ten years hence is •985. Find the probability that A and B alone will be living at the end of the tenth year.

MISCELLANEOUS EXAMPLES 379

71.Integrate

(i)

^—1

dx;(ii)

sin

—

cos

x dx.

4sinx+cosx

A closed circular cylinder of height h is to be inscribed in a given sphere of radius R. If the whole surface of the cylinder, including the base and the lid, is to be a maximum, prove that

— _ 7

₁

—

₅

Show that _dx{(I + 1/x)^x} = e/2x² approximately, if x is large.

A man constantly stakes a fixed proportion of his property in a fair wager in which if he wins he will increase his property by

/mth part, and if he loses he will decrease it by

/nth part. Show that in the long run he will lose.

The equation

2X — 20 = o

has a root between

2'4

and

2'5.

Determine the value of this root correct to four decimal places by a method of inverse interpolation.A large army consists of men between ages

and

40,

the number at age x being proportionate to

+ bc

where a,

and c are constants. If the numbers at ages

20,

and

are proportionate to 10o,

and

respectively, find, correct to one decimal place, the average age of the men in the army. Given

log

₁₀

'301,

log

₁₀

3 = '477

and log

₁₀

= •4343

Show that d^orlogx _ (—I)"n!(m+n—_{log x}(rn+r — I)!

en }

(m –

I)!

xm+n {

_r-O

r! (n — r)

The winner of a game is the one who first scores four points with the proviso that if two players score three points, the game conuntil one player has scored two points more than the other. A's skill is to B's as 2 : I. Find A's chance of winning the game if he owes one point and B receives a start of one point.

(i) Prove that e^x= (_E,) e^x.Qex (interval of differencing h). (ii) If u_s be a function of the form

₁

+ b

₂

₃

+ ...

to infinity,

show that it can be expressed in the form

_ b

₁

I —

(I —

(I –

x)3 + ....

380 ACTUARIAL MATHEMATICS

80.(i) If K/(y — u) = Kv/(t — x) = (1 + v²)}, where v = du WI and K is a constant, find the differential coefficient of y with respect to x. (ii) Given that be^aI^a = sin (y/a — c), find the value of

^Dy[I + (^Dy)²]/^D2y.

81.

The following data are available:

Age x3

374

475

35'3

⁶

33'

7227

2323

19'11

It is desired to obtain e₅₇with as little labour as possible, and it is suggested that 18^.71 would be a reasonable approximation. Do you agree with this? Give reasons.

From the above data, obtain a value for

₄

₁

82.Evaluate _fx²(log x)² dx and [sin^e x cos³x dx.

If u, be a function whose differences, when the increment of x is unity, are denoted by 8u_x, 8²u_,,, 8³u_x, ... and by Au,., A²u,, A³ux, ... when the increment of x is n ; then if 8²u_x, 8²u_s+1, ... are in geometric progression with common ratio q, show that

Aux —

(qn —

1) —

n (q —

^I)

(q —

¹)

84.Show that

n₁. 1⁴ + n₂.2⁴ - n₃.3⁴ + ... + n⁴= 2^n-4n (n + ^I) (n2 + 5n — 2).

A thin closed rectangular box is to have one edge n times the length of another edge and the volume is to be V. Prove that the least surface S is given by

= 54

(

ⁿ

1)2

Two men throw for a guinea, equal throws to divide the stake. A uses an ordinary die, but B uses a die marked 2, 3, 4, 5, 6, 6. Show that B thereby increases his expectation by 5/18ths.

Prove that if the polar coordinates of two points on the curve

f (B)

₁

, O)

and (r

₂

the area contained by the curve and the

two radii r

₁

and r

₂

z [ r

dO.

Hence prove that the whole area of the curve r

= a

cos

is a

Integrate r- ^dxaccording as

f— 1

1~x

k is a constant ;

x — a,

where

is independent of

1/k = x.

8
9.

v

MISCELLANEOUS EXAMPLES 381
log (I+e^x)-loge-fix
uate _x, o x log (x + V I + x²) and find the limit of
log x +logy when x and y each x+y-2

90. Explain the difference, if any, between A_x0,, and A_i,. Find ^a44^:51 , given
a₄₀.₅₀ =	10'894	^a40:55 =	9'79⁶	^a40:50 =	⁸'553
^a45:50 =	^10'59¹	^a45:55	9'5⁸3
^a50:50 =	10'059

91. A certain type of tag consists of a "bootlace" with a cylinder at one end into which the tag at the other end fits. If any number N of exactly similar tags be held in the middle so that the cylinder ends hang down at one side and the tag ends at the other:
(a) what is the chance that if, say, n tags be fitted into n cylinders at random, both ends have been chosen from the same " boot-lace," so that n loops are formed?

(b) if all the N tags be fitted into all the N cylinders, what is the chance that one large loop is formed?
92. Differentiate with respect to x:
x sin^-1 x log ae/x + -VI — x² log a/x + log {xi(I + — x²)}.

93. Three metal discs are numbered 1, 2, 4 respectively on one side: the other side of each disc is blank. The discs are tossed three times, and the numbers showing up are added. A is to win a stake from B if the total is 8 to 13 inclusive, while B wins if the total is less than 8 or more than 13. Find the odds in favour of B's winning.

94. Prove that, if y = cos (m cos-¹ nx), then
d²y (I - n2x2) _!j-^n2xd + m²n²y = o.
95. If interpolated values are found in the interval x = o to x = 1 from the values u_₁, 1~₀, u_1iu₂, by means of the formula
ux=Sn0+ ²( -^I)A²u-1 2
+ xu₁ + x2 _2-I) A2u0 (where = 1 - x), and in the next interval x = i to x = 2 by the corresponding formula based on the values u₀, u₁, u₂ and u₃, show that:
(1) The given values u₀, u₁ and u₂ will be reproduced by the interpolation.

382 ACTUARIAL MATHEMATICS

(2) The two interpolation curves have the same differential coefficient when x = I.

(3) The interpolated values for u₄ and u_i agree with those given by the ordinary third difference interpolation formula based on the same values of it, .

Given the following values:
u_, = 1000, u₁₀= 2609,
^Iro = ¹4⁰3, ¹15 = 34⁸7,
u, = ¹93¹,

complete the table for unit intervals from u_o to u_TOby the above formulae and calculate the value of the differential coefficient of the interpolated curves when x = 5.

96.Prove that the limit of the series

'Van

2) 1

2 (2n

2) + (n + 3) A/3 (2n + 3) + ...

Vn (3

ⁿ

)

when

n—>oo is

37r.

97.If yⁱ/^P + y ^{1 P}= 2x, prove that

(x2—I)D^+2y+(2n+ 1)xDn+Iy+(n2— p1)D_'z_y=o_.

98.

Solve the equation

_,x+1

_{x_I}

= u

Prove that

d {x^m-2(I — 2x²y^'+1} = Ax^m-3(I — 2x²)^P + Bx^m-1(I — 2x²)^P,

where

and

are constants.

Find

and

and hence show that

(I —

can be expressed

in the form

al — {

f (x).

(T —

)

^''}

where

= J

1(I —

)

and a is a constant.

Too. If

is a first approximation to a root of an equation

f (x) = o,

show that a —

f (a); f

(a)

is likely to be a better approximation.

Apply the method to determine, correct to three places of decimals, the root of

⁴

—

8x — 6o = o

which is nearly equal to

lox. If in an examination six men are bracketed, the extreme difbetween their marks being 6, find the chance that they have all obtained different marks.102. The xth term of the series I, 2, 17, 72, 243, 754, ... is of the form a + bx + c^x+ d^x. Determine a, b, c, d and find the sum of n terms of the series.