a2 math c3 done

8/18/2019 A2 Math C3 done

1/53

Chapter 3

Continuous probability distributions

In this chapter you will learn

• about the probability density function of a continuous random variable X

• how to find a probability by calculating the area under the graph of y = f ( x)

• how to find the median and quartiles of X • how to find the mean and standard deviation of X

CONTINUOUS RANDOM VARIABLES

You have already seen in S chapter ! that a discrete random variable is a variable that

can ta"e individual values each with a given probability# $or e%ample& the probabilitydistribution of '& the score on a fair cubical die is as follows

So& for instance& ( X = 3) =

* &

+y contrast& a continuous random variable cannot ta"e precise values but can be defined

only within a specified interval# $or e%ample& when a boy,s height is given as -* cm&measured to the nearest cm& this means that the height could be anywhere in the interval

-.#. cm / height -*#. cm# Continuous variables are associated with measurements of˂

characteristics such as time& mass or length#

Probabilit densit !"nction

continuous random variable 'is defined by its #robabilit densit !"nction f& and it can be illustrated by the graph of y = f(%)& for e%ample

0ote that a thic"er line is drawn on the %1a%is outside the interval / x /- to show that y

= 2 for these values of %#

$indin% #robabilities

If a continuous random variable ' has probability density function f then the #robabilit

that % lies in the interval a / x / b is given by the area under the graph of y = f(%)

between x = a and x = b#his area can be found by inte%ration& or sometimes by geometry#

0ote


2/53

• Since probabilities cannot be negative& the graph of y = f(%) never goes below the

%1a%is#

• Since the total probability is & the total area under the graph is #

$or a contin"o"s random variable X with #robabilit densit !"nction given by f(%)

for a ≤ x ≤ b&

( x/ X / x-) =

-

( )

x

x

f x dx∫

Since the total area under the curve is &

( ) ( ) b

all x a

f x dx f x dx= =∫ ∫

0ote his compares with discrete random variables& where

n

i

i

p=

=∑ (S Chapter !)

0ote that this is different from calculating probabilities for discrete variables (such as

variables having a binomial or a oisson distribution)#$or discrete random variables& ( X = 3)& for e%ample& has a definite value which is

usually not 4ero# his is different for continuous random variables such as time& fore%ample& where a measurement of 3 seconds correct to the nearest tenth of a second could be anywhere in the interval -#5. seconds / ' 6 3#2. seconds# his interval becomes

narrower and narrower as you try to approach the instant of time of 3 seconds and the

probability that X ta"es the e%act value of 3 is 4ero#

In fact& for any continuous random variable '

( X = k ) = 2& where k is any constant#


3/53

You will recall that for discrete random variables& ( X 6 3) is not the same as ( X / 3)#

7owever& for continuous variables it is not possible to distinguish between ( X 6 3) and

(' / 3)& nor between (' 8 3) and (' 9 3)#In general& if ' is a continuous random variable it is not possible to distinguish between

the following probabilities& but if X is discrete they would be different

( xl 6 X 6 x-)& ( xl / X 6 x-)& ( xl 6 X / x-)& ( x / X / x-)his difference between discrete and continuous random variables e%plains the need for a

continuity correction when using the normal distribution as an appro%imation to the

binomial distribution (S chapter *) or oisson distribution (S- chapter )#

Inte%ration note

In the e%amination you may be required to integrate functions described in the integration

sections of and 3# 7owever& so that you can study this chapter before all theintegration methods have been covered& 3 functions occur only in the miscellaneous

e%amples and :i%ed ;%ercise 3 at the end of the chapter#

;%ample 3#

he continuous random variable ' has probability density function given byf ( x )={k x

2,∧1≤ x ≤4

0,∧otherwise where k is a constant#

(i)


4/53

;%ample 3#-

he random variable ' denotes the mass& in "ilograms& of a substance produced in anindustrial process# he probability density function of ' is given by

f ( x )={ 1

36 x (6− x) ,0≤ x ≤6

0,∧otherwise

Calculate the probability that more than . "g of the substance is produced in the

industrial process#


5/53

;%ample 3#3

he error& in grams& made by a set of weighing scales may be modelled by the randomvariable ' with probability density function given by

f ( x )={k ,−¿3≤ x ≤70,∧otherwise where k is a constant#

(i) $ind the value of "#

(ii) $ind the probability that an error is positive#(iii) iven that an error is positive& find the probability that the error is less

than ! g#

(iv)$ind the probability that the ma%nit"de of an error is less than - g#


6/53

;%ample 3#!

he continuous random variable has probability density function given by

f ( x )=

{ k

t 4 , t ≥1

0,∧otherwise where k is a constant#

(i) $ind the value of k #

(ii) Calculate (T 6 #.)#


7/53


8/53

;%ercise 3a

# he continuous random variable X has probability density function given by

f ( x )={k x2

,∧0≤ x ≤20 ,∧otherwise

where k is a constant#


(ii) $ind ( X 9 )#(iii) $ind (2#. / X / #.)#

-# he continuous random variable X has probability density function given by

f ( x )={k ,∧−2≤ x ≤30 ,∧otherwisewhere k is a constant#

(i) S"etch the probability density function of X #

(ii) $ind the value of k #

(iii) $ind (1#* / X / -#)#(iv)$ind (1-#. 6 X 6 -#.)#

0ote (1-#. 6 ' 6 -#.) can be written ( | x| 6 -#.)#(v)

(a) $ind (' 8 )#(b) iven than X is greater than & find the probability that X is less than #.

3# he continuous random variable X has probability density function given by

f ( x )={k (4− x) ,∧1≤ x ≤30,∧otherwisewhere k is a constant#


(ii) S"etch the probability density function of X #(iii) $ind ( X 8 -)#

(iv)$ind (#- / X / -#!)#

!# he continuous random variable X has probability density function given by

f ( x )={k ( x+2)2

,∧0≤ x ≤20,∧otherwise



9/53


(ii) $ind (2 / X /) and hence find ( X 8 )

.# he continuous random variable X has probability density function given by

f ( x )={k x3

,∧0≤ x ≤ c0,∧otherwise

where k and c are constants#

iven that P( X ≤ 12 )= 116 is& find c and k.

*# he continuous random variable ' has probability density function given by

f ( x )={k x ,∧0≤ x ≤40,∧otherwisewhere k is a constant#


(ii) S"etch the probability density function of X #(iii) $ind ( / X / -#.)#

># he delay& in hours& of a flight from Chicago can be modelled by the continuous

random variable ' with probability density function given by

f ( x )={ 1

50 (10− x ) ,∧0≤ x ≤10

0,∧otherwise

(i) $ind the probability that the delay will be less than ! h#

(ii) $ind the probability that the delay will be between - h and * h#

?# he continuous random variable X has probability density function given by


10/53

f ( x )={ k ,∧a ≤ x ≤100 ,∧otherwisewhere k is a constant#It is "nown that (. 6 X 6 ?) = 2#. $ind the values of k and a#

5# he continuous random variable ' has probability density function given by

f ( x )={k √ x ,∧1≤ x ≤90,∧otherwisewhere k is a constant#


(ii) $ind (! / X / 5)#

2# he continuous random variable X has probability density function given by

f ( x )={ k

x2

,∧1≤ x ≤3

0,∧otherwisewhere k is a constant#


(ii) $ind ( X / -)#

# he continuous random variable X has probability density function given by

f ( x )={ 2

x3 ,∧ x ≥1

0,∧otherwise


11/53

$ind ( X 8 -)#

-# ;%plain& with a reason& whether each of these functions could be the probability

density function of a random variable X #

3# he random variable X has probability density function given by

f ( x )={k ( x2−4 ) ,∧0≤ x ≤20,∧otherwise


$ind the value of k #


12/53

!# he random variable X has probability density function given by

f ( x )={ k

x4

,∧ x ≥2

0 ,∧otherwisewhere k is a constant#

(i) $ind the value of k #(ii) $ind the probability that X is greater than 3#

.# he random variable ' has probability density function given by

f ( x )={a (b− x ) ,∧0≤ x ≤ b0 ,∧otherwisewhere a and b are constants#

(i) Show that a= 2

b2

(ii) iven that a=1

8

find (- / X / 3)#

MEDIAN AND &UARTILESConsider the continuous variable '& defined by its probability density function f for

a / x / b#

Medianhe median of the distribution of X is defined as the value which divides the area under

y = f ( x) in half#

&"artileshe quartiles of the distribution of X are the values which& together with the median&

divide the area under y = f ( x) into quarters#

@emember that


13/53

interquartile range = upper quartile 1 lower quartile

= q3 1 q

;%ample 3#.

he continuous random variable ' has probability density function given by

f ( x )={18

x ,∧0≤ x ≤4

0,∧otherwise (i) $ind the median of X # (ii) $ind the interquartile range#


14/53

0oteIn ;%ample 3#.& since y = f( x) is a straight line& the median and quartiles could be found

using geometry# $or e%ample& to find the median


15/53

;%ample 3#*


f ( x )=

{ 8

3 x2

,∧1≤ x ≤2

0,∧otherwise$ind the median of X #

;%ample 3#>



16/53

f ( x )={1

8(4− x) ,∧0≤ x ≤ 4

0,∧otherwise(i) $ind ( X 8 -) and deduce the value of the upper quartile#

(ii) $ind the median of X #


17/53

So& median = #> (3 s#f#)


18/53

0ote As y = f( x) is a straight line& the median could also be found using geometry#

Usin% %eometr'

Bhen x = m& y=1

8(4−m)

Area of shaded triangle ¿1

2 × (4−m) ×

1

8(4−m )=

1

16(4−m)2

+ut& since ( X 8 m) = 2#.& area of shaded triangle = 2#.#

So1

16

(4−m)2=0.5

(4−m)2=8Square root both sides:

4−m=±√ 8m=! √ 8 =*#?-?## (reDect) or m=!1 √ 8 = #>###So& median = #> (3 s#f#)& as above#

E(ercise )b

# he continuous random variable ' has probability density function given by

f ( x )=

{3

8

x2,∧0≤ x ≤2

0,∧otherwise(i) $ind the median#(ii) $ind the upper and lower quartiles#

(iii) $ind the interquartile range#


19/53

-# he continuous random variable ' has probability density function given by

f ( x )={1−1

4 , 1≤ x ≤3

0,∧otherwise(i) $ind the median#

(ii) iven that ' is greater than the median& find the probability that ' is less than theupper quartile


f ( x )={ 3

2 x2

,∧1≤ x ≤3

0,∧otherwise(i) $ind the median#

(ii) Show that the upper quartile is - and find the lower quartile#

!# he continuous random variable ' has probability density function given by

f ( x )={1+ x

k ,∧1≤ x ≤3


(i) $ind the value of "#(ii) S"etch y = f(%)#

(iii) $ind the median

(iv)$ind the probability that e%actly ! out of * random observations of X have valuesless than the lower quartile#

.# he continuous random variable ' has probability density function given by

f ( x )={ 1

18(3+ x ) ,∧−3≤ x ≤3

0,∧otherwise$ind the lower quartile of '#


f ( x )={3

8(1+ x2),∧−1≤ x ≤1

0,∧otherwise(i) S"etch the probability density function#

(ii) State the value of the median#

># he continuous random variable ' has probability density function given by

f ( x )={ 3 x−1

,∧ x ≥30,∧otherwise

(i) $ind the median#

(ii) Show that ( X 8 -) = 2#-. and hence state the value of the upper quartile#(iii) $ind the interquartile range#


20/53

?# he continuous random variable ' has probability density function given by

f ( x )={32

3 x

−3,∧2≤ x ≤4

0,∧otherwise$ind the median of '#

MEAN AND VARIANCEMean *e(#ectation+ o! ,

he mean of '& also called the e%pectation or e%pected value of '& is written ;(') and isdenoted by µ#

$or a continuous random variable X with probability density function f( x)& the mean (or

e%pectation) of X is given by

μ= E ( X )=∫all x xf ( x) dx he formula for ;(') is given in the e%amination#

0ote Compare this with discrete random variables& where E = ;(') =

n

i i

i

x p=

∑ (S Chapter !)

;%ample 3#?

he random variable X has probability density function given by

f ( x )={ 1

18(6− x) ,∧0≤ x ≤6

0,∧otherwise$ind ;( X )#


21/53

;%ample 3#5

he random variable ' has probability density function given by

f ( x )=

{

1

9 x

2,∧0≤ x ≤3

0,∧otherwisehe mean of X is and the median of X is m#

(i) $ind µ#

(ii) $ind ( X 6 µ)#

(iii) $ind the probability that X lies between µ and m#


22/53

;%ample 3#2

he continuous random variable ' has probability density function f& where 2 / x / 2he diagram shows the graph of y = f( x)#

(i) $ind the value of k #(ii) $ind ;( X )#


23/53

;%ample 3#he continuous random variable & has probability density function given by

f ( t )={ k

t 3 ,∧t ≥3



(ii) $ind ;()


24/53


25/53

;%ample 3#-

At a town centre car par" the length of stay in hours is denoted by the random variable X &

which has probability density function given by

f ( x )=

{k x

−32 ,∧1≤ x ≤9

0,∧otherwise


(i) Interpret the inequalities / % / 5 in the definition of f( x) in the conte%t of thequestion#

(ii) Show that k =3

4#

(iii) Calculate the mean length of stay

he charge for a Iength of stay of % hours is ( 1−e− x ) dollars#(iv)$ind the length of stay for a charge to be at least 2#>. dollars#

(v) $ind the probability of the charge being at least 2#>. dollars#Cambridge aper > F> 02*


26/53


27/53

;%ample 3#3


f ( x )={ax+b ,∧0≤ x ≤20,∧otherwise

where a and b are constants#

It is given that ;(') =16

15# $ind the value of a and the value of b.


28/53

Note abo"t smmetr

If the probability density function f is defined for a / % / b and the graph of y = f( x) has aline of symmetry in this interval& then the mean is the midpoint of the interval

µ = ;( X ) = 1

2(a b)

$or e%ample& consider the random variable ' defined in ;%ample 3#-#


29/53

f ( x )={ 1

36 x (6− x) ,∧0≤ x ≤6

0,∧otherwise

;%ercise 3c# he continuous random variable ' has probability density function given by

f ( x )=

{

1

16 x ,∧2≤ x ≤6

0,∧otherwise$ind ;(')#


f ( x )={ x+320

,∧0≤ x ≤4

0,∧otherwise$ind the value of ;(')#


f ( x )=

{34 ( x

2+1) ,∧0≤ x ≤1

0,∧otherwise

$ind the value of ;(')#

!# he continuous random variable Y has probability density function given by


30/53

f ( y )={ 3

14 √ y ,∧1≤ y ≤4

0,∧otherwise

$ind the value of ;(Y)#


f ( x )={3

4 x (2− x) ,∧0≤ x ≤2

0,∧otherwise(i) S"etch the probability density function of '#

(ii) $ind the mean value of X.


f ( x )=

{1

4 x3

,∧0≤ x ≤2

0,∧otherwise(i) $ind ;(')#

(ii) $ind (' 6 ;('))#

(iii) Is the mean of ' less than or greater than the median of 'G Hustify youranswer#

># he random variable ' denotes the lifetime& in years& of a particular type of light bulb# he probability density function of ' is given by

f ( x )=

{kx (5− x) ,∧0≤ x ≤5

0,∧otherwise(i) Show that " =

6

125

(ii) wo light bulbs are selected at random#

(iii) $ind the probability that both light bulbs last longer than the mean lifetimeof this type of light bulb#


f ( x )={ 5

32 x

4,∧0≤ x ≤2

0,∧othe rwise

(i) $ind ;(')#(ii) $ind the median m#

(iii) $ind the probability that a random observation of ' lies between the mean


f ( x )={ k ,∧a ≤ x ≤ b0,∧otherwisewhere k & a and b are positive constants#

(i) ;%press " in terms of a and b#


31/53

he mean of X is ? and the interquartile range is *#

(i) $ind the values of a& b and k #

(ii) $ind (' 8 -)#

2# he continuous random variable has probability density function given by

f ( x )={k t 4 (4− x) ,∧t ≥1


(i) Show that k = 3#

(ii) $ind ;(>)#


f ( x )=

{

k 3√ x

,∧1≤ x ≤8


(i) Show that " =2

9

(ii) $ind ;(')#


f ( x )={ p−qx ,∧0≤ x ≤20,∧otherwisewhere p and q are constants#

(i) Show that - p 1 -q = #

(ii) iven that the mean of ' is 23

&

(a) form a second equation in p and q&

(b) find the value of p and the value of q#

Variance o! ,$or continuous random variables& the variance of '& denoted by -& is defined as follows

σ 2=Var ( X )=∫

all x

❑

( x− μ )2 f ( x )dx where µ = ;( X )

7owever& this formula can be complicated to wor" with& so an alternative version derived

by e%panding the brac"et is usually used# his is shown below#

$or a continuous random variable ' with probability density function f& the variance of'& Jar(')& is denoted by -& where

σ 2=Var ( X )=∫

all x

❑

x2

f ( x ) dx− μ2 where μ = ∫all x

❑

xf ( x ) dx

his compares with the two versions of the variance formula for discrete random variables where


32/53

- - -

( ) ( )n n

i i i i

i i

Var X x p x p µ µ = =

= − = −∑ ∑

Kn the formulae list provided in the e%amination& the e%pectation and variance formulaeare given as follows

E ( X )=

∫ xf ( x ) dx Var ( X )=∫ x2 f ( x) dx− { E ( X )}2

;%ample 3#!he random variable ' has probability density function given by

f ( x )={3 xk


where k is a positive constant#(i) $ind the value of "#

(ii) Show that the mean& µ& of ' is 2#>.

(iii) Show that the standard deviation& & is 2#53*& correct to ! significantfigures#

(iv)$ind ( μ−σ ≤ X ≤ μ+σ )#


33/53


34/53

;%ample 3#.A continuous random variable ' has probability density function given by

f ( x )={3(1− x )2


$ind

(i) (' 8 2#.)#(ii) he mean and variance of '#

Cambridge aper > F* 02!


35/53


36/53

;%ample 3#*

A continuous random variable ' has probability density function given by

f ( x )={ 1

c ,∧0≤ x ≤ c

0,∧otherwise(i) State the value of ;(') in terms of c#

(ii) $ind Jar(') in terms of c#

(iii) If c = *& find the standard deviation of '#


37/53

;%ercise 3d


f ( x )={2

5 x ,∧2≤ x ≤3


(ii) $ind Jar(')#


f ( x )={3

7 x

2,∧1≤ x ≤2


(ii) $ind Jar(')#

(iii) $ind the standard deviation of '#


f ( x )={k ,∧−2≤ x ≤30,∧otherwise


(ii) State the value of ;(')#(iii) $ind Jar(')#

(iv)$ind the standard deviation of '#

!# he continuous random variable ' has mean p and standard deviation a1# he

probability density function of ' is given by


38/53

f ( x )={ 1

32(8− x) ,∧0≤ x ≤8

0,∧otherwise

(i) Show that E = 22

3

(ii) Show that σ 2=3 5

9

(iii) $ind ( X ≥ μ+σ )#


f ( x )={ 3

16(4− x2) ,∧0≤ x ≤2

0,∧otherwise(i) Show that the mean of ' is 2#>.

(ii) $ind the variance of '#

*# he continuous random variable has probability density function given by

f ( x1 )={5

6(t 4+1),∧0≤ t ≤1

0,∧otherwise

(i) Show that ;() =5

9

(ii) $ind the variance of

># he mass& in "ilograms& of metal e%tracted from 2g of ore from a certain mine is

a continuous random variable ' with probability density function

f ( x )={34

x(2− x)2 ,∧0≤ x ≤2

0,∧otherwise(i) Show that the mean mass is 2#? "g#

(ii) $ind the standard deviation of the mass of metal e%tracted#

?# he continuous random variable has probability density function given by

f ( t )={ k

√ t ,∧1≤t ≤ 4



(ii) $ind the standard deviation of #

5# he continuous random variable Y has probability density function given by

f ( y )={ a

y4 ,∧ y ≥2

0,∧otherwisewhere a is a constant#


39/53

(i) Show that a = -!#

(ii) $ind ;( )#

(iii) $ind Jar( )#

MISCELLANEOUS -OR.ED E,AMPLES/

he e%amples in this section include 3 integration methods#

;%ample 3>

he random variable denotes the time in seconds for which a firewor" burns before

e%ploding# he probability density function for is given by

f ( x t )={k e0.2 t

,∧0≤t ≤50,∧otherwise


(i) Show that " =1

5 (e−1)(ii) S"etch the probability density function#

(iii) ?2L of firewor"s burn for longer than a certain time before they e%plode#

$ind this time#Cambridge aper > F. H2(>)


40/53

;%ample 3#?

he continuous random variable ' has probability density function given by f(%)

f ( x )=

{kcosx ,∧0≤ x ≤

1

4


(i) Show that " = √ 2 #(ii) $ind (' 8 2#!)# &(iii) $ind the upper quartile of '#

(iv)$ind the probability that e%actly 3 out of . random observations of X have values

greater than the upper quartile#

Cambridge aper > F. 025(>)


41/53

= 2#2?>5 (3 s#f# )


42/53

;%ample 3#5

If Msha is stung by a bee she always develops an allergic reaction# he time ta"en in

minutes for Msha to develop the reaction can be modelled using the probability densityfunction given by

f ( x )=

{ k

t +1,∧0≤t ≤ 4


(i) Show that k =1

ln5

(ii) $ind the probability that it ta"es more than 3 minutes for Msha to develop the

reaction#

(iii) $ind the median time for Msha to develop a reaction#Cambridge aper > F> H2?


43/53

= 2#3?*N


44/53

;%ample 3#-2

he lifetime& % years& of the power light on a free4er& which is left on continuously& can be

modelled by the continuous random variable with density function given by

f ( x )={k e

−3 x

,∧ x>00,∧otherwise


(i) Show that k = 3#(ii) $ind the lower quartile#

(iii) $ind the mean lifetime#

Cambridge aper > F> 0K3


45/53

" = 3


46/53

E(ercise )e


f ( x )=

{32

3

x−3

,∧2≤ x ≤4


(ii) $ind Jar())#

-# he random variable has probability density function given by

f ( t )={k et

,∧0≤ t ≤10,∧otherwise

where " is a constant#


47/53

(i) Show that k = 1

e−1(ii) $ind the median of #(iii) $ind the mean of #

3# he continuous random variable ' has probability density function given by )f"f ( x )={ksi!x ,∧0≤ x ≤ 0,∧otherwise where k is a constant#

(i) Show that " =1

2@emember to wor" in radians#

(ii) +y considering a s"etch of the probability density function& state the value of the

mean of '#

(iii)

(a) Show that (' 61

3 ) =

1

4

(b) $ind (' 62

3

¿

(c) $ind the interquartile range#

!# he continuous random variable ' has probability density function given by

f ( x )={k ( x−1 )6



Msing the substitution u = % 1 (i) show that " = >

(ii) find the mean of '

(iii) find the median of '#


f ( x )={ksec2 x ,∧0≤ x ≤ 1

4


(i) Show that k = @emember to wor" in radians#

(ii) $ind the median of '#(iii) $ind the interquartile range#



48/53

f ( x )={ k

x ,∧1≤ x ≤4

0,∧otherwisewhere " is a constant#

(i) Show that " =1

2ln2

(ii) $ind ;( X )#

(iii) $ind Jar( X )#

(iv)Show that the median is -#

(v) Show that the lower quartile is v and find the upper quartile#

># he time& in years& that ;duardo "eeps his car before replacing it with a new onecan be modelled by a continuous random variable with probability density

function given by

f ( x )={ 14 e−14

t

,∧t >0

0,∧otherwise

(i) $ind the probability that he "eeps his car less than year before replacing it#

(ii) $ind the probability that he "eeps his car for more than - years before replacing it#(iii) $ind the mean length of time ;duardo"eeps his car before replacing it#


f ( x )=

{ k x

2

x3

−1

x ,1.5≤ x ≤3.5



49/53

$ind the value of "#


f ( x )=

{

k

( x−1)( x−2),∧3≤ x ≤5


$ind the value of k

2# he continuous random variable ' has probability density function given by f(%)

f ( t )={kx e2 x ,0≤ x ≤0.5



(ii) Show that ;(') =12

e−1 @ecall Integration by parts

# he continuous random variable has probability density function given by

f ( x )={ k

t +1,1≤ t ≤3

0,∧otherwis ewhere k is a constant#

(i) Show that k =1

ln 2

(ii) +y using the substitution u = t & or otherwise& find ;(T )#

S"mmar$or a contin"o"s random variable ' with probability density function defined by f( x)for a / x / b

robabilities are given by areas under the curve

( x / X / x-) = ∫ x1

x2

f ( x ) dx


50/53

he total area under the curve is

∫all x

❑

f ( x ) dx=1

f(%) 9 2 for all values of x& so the graph of y = f( x) never goes below the %1a%is#

Median and 0"artiles

Mean and variance

:i%ed ;%ercise 3

# he random variable ' has probability density function given by

f ( x )={kx ,∧0≤ x ≤20,∧otherwisewhere " is a constant#

(i) $ind the value of "#(ii) $ind the median of '#

(iii) $ind the mean of '#

-# he random variable ' has probability density function given by

f ( x )={4 xk


where " is a positive constant#

(i) Show that " = 3

(ii) Show that the mean of ' is 2#? and find the variance of '#

(iii) $ind the upper quartile of '#(iv)$ind the interquartile range of '#

Cambridge aper > F. H2*

3# A continuous random variable ' has probability density function given by

f ( x )={1

6 x ,∧2≤ x ≤4


(ii) $ind the median of '#


51/53

(iii) wo independent values of ' are chosen at random# $ind the probability

that both these values are greater than 3#

Cambridge aper > F. 02(>3)

!# he random variable ' has probability density function given byf ( x )={k x (6− x)

2,∧0≤ x ≤6


(i) Show that " =1

108

(ii) $ind ;(')#(iii) $ind the standard deviation of '

.# he random variable ' denotes the number of hours of cloud cover per day at aweather forecasting centre# he probability density function of 'is given by

f ( x )={( x−18 )2

k x ,∧0≤ x ≤24

0,∧otherwise


(i) Show that k = -2*#

(ii) Kn how many days in a year of 3*. days can the centre e%pect to have less than -hours of cloud coverG

(iii) $ind the mean number of hours of cloud O cover per day#

Cambridge aper > F> H2.


f ( x )={3

4( x2−1) ,∧1≤ x ≤2

0,∧otherwise(i) S"etch the probability density function of '#

(ii) Show that the mean& μ& of ' is #*?>.#(iii) Show that the standard deviation& & of ' is 2#--??& correct to ! decimal

places#

(iv)$ind ( / X / µ ! )#Cambridge aper > F> H2>

># he time T & in minutes& that 7elen has to wait for the bus when she is travelling towor" has probability density function given by

f ( t )={ k ,∧0≤ t ≤100,∧otherwise(i) Bhat is the longest time that 7elen has to wait for the busG(ii) State the mean time she has to wait for the bus#

(iii) $ind the standard deviation of the time she has to wait for the bus#

(iv)$ind the probability that the time she has to wait is more than standard deviationaway from the mean#


52/53


f ( x )={a+bx ,∧0≤ x ≤10,∧otherwise(i) Show that -a b = -#

he median of X is 2#*#(ii) $ind a second equation in a and b and hence find the values of a and b#

5# he time in hours ta"en for clothes to dry can be modelled by the continuous

random variable with probability density function given by

f ( x )={ k √ t ,1≤ t ≤40,∧otherwisewhere " is a constant#

(i) Show that " = ! #(ii) $ind the mean time ta"en for clothes to dry#

(iii) $ind the median time ta"en for clothes to dry#

(iv)$ind the probability that the time ta"en for clothes to dry is between the mean

time and the median time#Cambridge aper > F> 02?

2# he time in minutes ta"en by candidates to answer a question in an e%amination

has probability density function given by

f ( x )={k (6t −t 2),∧3≤t ≤6


(i) Show that " = 1

18 #

(ii) $ind the mean time#

(iii) $ind the probability that a candidate& chosen at random& ta"es longer than. minutes to answer the question#

(iv)Is the upper quartile of the times greater than . minutes& equal to . minutes or less

than . minutesG ive a reason for your answer#

Cambridge aper > F. 325(>)

# he average speed of a bus& x "m h1& on a certain Dourney is a continuous random

variable X with probability density function given by

f ( x )=

{

k

x2 ,∧20≤ x ≤28

0,∧otherwise(i) Show that#" = >2#(ii) $ind ;(')#

(iii) $ind (' 6(iv)7ence determine whether the mean is greater or less than the median#

Cambridge aper > F* 0K-



53/53

f ( x )={a x2+bx ,∧0≤ x ≤20,∧otherwise

where a and b are constants#he mean of ' is #-.

(i) Show that b = ! and find the value of a#

(ii) $ind the variance of '#(iii) Jerify that the median of ' is appro%imately #3

3# he lifetime t & in hours& of a certain type of electrical component can be modelled

by a continuous random variable with density function given by

f ( x )={0.05e−0.05t

,∧t >00,∧otherwise

(i) A component is chosen at random from the production line#

$ind the probability that

(a) the component will fail within - hours

(b) the component will last longer than * hours#(ii) $ind the median lifetime of a component of this type#

(iii) Show that the mean lifetime of a component of this type is -2 hours#

!# he continuous random variable X has probability density function given by

f ( x )={kcosx,∧0≤ x ≤ 1

2


(i) $ind the value of k.(ii) $ind ;( X )#

(iii) $ind the median of X

.# he continuous random variable X has probability density function given by

f ( x )={k e2 x



(i) Show that k =2

e8−1

(ii) $ind the mean of X #

*# he continuous random variable X has probability density function given by

f ( x )={ k

x3 ,∧2≤ x ≤3

0,∧otherwise

a2 math c3 done

Documents