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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
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• 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#
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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)
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;%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#
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;%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#
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;%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 #.)#
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;%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#
(i) $ind the value of k #
(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#
(i) $ind the value of k #
(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
where k is a constant#
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(i) $ind the value of k #
(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#
(i) $ind the value of k #
(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
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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#
(i) $ind the value of k #
(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#
(i) $ind the value of k #
(ii) $ind ( X / -)#
# he continuous random variable X has probability density function given by
f ( x )={ 2
x3 ,∧ x ≥1
0,∧otherwise
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$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
where k is a constant#
$ind the value of k #
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!# 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
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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#
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0oteIn ;%ample 3#.& since y = f( x) is a straight line& the median and quartiles could be found
using geometry# $or e%ample& to find the median
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;%ample 3#*
he continuous random variable ' has probability density function given by
f ( x )=
{ 8
3 x2
,∧1≤ x ≤2
0,∧otherwise$ind the median of X #
;%ample 3#>
he continuous random variable ' has probability density function given by
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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 #
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So& median = #> (3 s#f#)
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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#
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-# 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
3# he continuous random variable ' has probability density function given by
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
0,∧otherwisewhere k is a constant#
(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 '#
*# he continuous random variable ' has probability density function given by
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) $ind the interquartile range#
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?# he continuous random variable ' has probability density function given by
f ( x )={32
3 x
−3,∧2≤ x ≤4
0,∧otherwise$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 )#
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;%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#
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;%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 )#
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;%ample 3#he continuous random variable & has probability density function given by
f ( t )={ k
t 3 ,∧t ≥3
0,∧otherwisewhere k is a constant#
(i) $ind the value of "#
(ii) $ind ;()
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;%ample 3#-
At a town centre car par" the length of stay in hours is denoted by the random variable X &
which has probability density function given by
f ( x )=
{k x
−32 ,∧1≤ x ≤9
0,∧otherwise
where k is a constant#
(i) Interpret the inequalities / % / 5 in the definition of f( x) in the conte%t of thequestion#
(ii) Show that k =3
4#
(iii) Calculate the mean length of stay
he charge for a Iength of stay of % hours is ( 1−e− x ) dollars#(iv)$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*
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;%ample 3#3
he continuous random variable ' has probability density function given by
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.
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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)
$or e%ample& consider the random variable ' defined in ;%ample 3#-#
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f ( x )={ 1
36 x (6− x) ,∧0≤ x ≤6
0,∧otherwise
;%ercise 3c# he continuous random variable ' has probability density function given by
f ( x )=
{
1
16 x ,∧2≤ x ≤6
0,∧otherwise$ind ;(')#
-# he continuous random variable ' has probability density function given by
f ( x )={ x+320
,∧0≤ x ≤4
0,∧otherwise$ind the value of ;(')#
3# he continuous random variable ' has probability density function given by
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
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f ( y )={ 3
14 √ y ,∧1≤ y ≤4
0,∧otherwise
$ind the value of ;(Y)#
.# he continuous random variable ' has probability density function given by
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.
*# he continuous random variable ' has probability density function given by
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#
?# he continuous random variable ' has probability density function given by
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
5# he continuous random variable ' has probability density function given by
f ( x )={ k ,∧a ≤ x ≤ b0,∧otherwisewhere k & a and b are positive constants#
(i) ;%press " in terms of a and b#
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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
0,∧otherwisewhere k is a constant#
(i) Show that k = 3#
(ii) $ind ;(>)#
# he continuous random variable ' has probability density function given by
f ( x )=
{
k 3√ x
,∧1≤ x ≤8
0,∧otherwisewhere k is a constant#
(i) Show that " =2
9
(ii) $ind ;(')#
-# he continuous random variable ' has probability density function given by
f ( x )={ p−qx ,∧0≤ x ≤20,∧otherwisewhere p and q are constants#
(i) Show that - p 1 -q = #
(ii) iven that the mean of ' is 23
&
(a) form a second equation in p and q&
(b) find the value of p and the value of q#
Variance o! ,$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
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- - -
( ) ( )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
,∧0≤ x ≤10,∧otherwise
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 )#
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;%ample 3#.A continuous random variable ' has probability density function given by
f ( x )={3(1− x )2
,∧0≤ x ≤10,∧otherwise
$ind
(i) (' 8 2#.)#(ii) he mean and variance of '#
Cambridge aper > F* 02!
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;%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 '#
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;%ercise 3d
# he continuous random variable ' has probability density function given by
f ( x )={2
5 x ,∧2≤ x ≤3
0,∧otherwise(i) $ind ;(')#
(ii) $ind Jar(')#
-# he continuous random variable ' has probability density function given by
f ( x )={3
7 x
2,∧1≤ x ≤2
0,∧otherwise(i) $ind ;(')#
(ii) $ind Jar(')#
(iii) $ind the standard deviation of '#
3# he continuous random variable ' has probability density function given by
f ( x )={k ,∧−2≤ x ≤30,∧otherwise
(i) $ind the value of k #
(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
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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 )#
.# he continuous random variable ' has probability density function given by
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
0,∧otherwisewhere k is a constant#
(i) $ind the value of "#
(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#
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(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
where k is a constant#
(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(>)
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;%ample 3#?
he continuous random variable ' has probability density function given by f(%)
f ( x )=
{kcosx ,∧0≤ x ≤
1
4
0,∧otherwisewhere k is a constant#
(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#
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= 2#2?>5 (3 s#f# )
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;%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
0,∧otherwisewhere k is a constant#
(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?
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= 2#3?*N
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;%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
where k is a constant#
(i) Show that k = 3#(ii) $ind the lower quartile#
(iii) $ind the mean lifetime#
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" = 3
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E(ercise )e
# he continuous random variable ' has probability density function given by
f ( x )=
{32
3
x−3
,∧2≤ x ≤4
0,∧otherwise(i) $ind ;(')#
(ii) $ind Jar())#
-# he random variable has probability density function given by
f ( t )={k et
,∧0≤ t ≤10,∧otherwise
where " is a constant#
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(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
,∧1≤ x ≤20,∧otherwise
where k is a constant#
Msing the substitution u = % 1 (i) show that " = >
(ii) find the mean of '
(iii) find the median of '#
.# he continuous random variable ' has probability density function given by
f ( x )={ksec2 x ,∧0≤ x ≤ 1
4
0,∧otherwisewhere k is a constant#
(i) Show that k = @emember to wor" in radians#
(ii) $ind the median of '#(iii) $ind the interquartile range#
*# he continuous random variable ' has probability density function given by
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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) $ind the mean length of time ;duardo"eeps his car before replacing it#
?# he continuous random variable ' has probability density function given by
f ( x )=
{ k x
2
x3
−1
x ,1.5≤ x ≤3.5
0,∧otherwisewhere " is a constant#
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$ind the value of "#
5# he continuous random variable ' has probability density function given by
f ( x )=
{
k
( x−1)( x−2),∧3≤ x ≤5
0,∧otherwisewhere k is a constant#
$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
0,∧otherwisewhere k is a constant#
(i) $ind the value of "#
(ii) Show that ;(') =12
e−1 @ecall Integration by parts
# he continuous random variable has probability density function given by
f ( x )={ k
t +1,1≤ t ≤3
0,∧otherwis ewhere k is a constant#
(i) Show that k =1
ln 2
(ii) +y using the substitution u = t & or otherwise& find ;(T )#
S"mmar$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
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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
,∧0≤ x ≤10,∧otherwise
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
0,∧otherwise(i) $ind ;(')#
(ii) $ind the median of '#
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(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
0,∧otherwisewhere " is a constant#
(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
where k is a constant#
(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) $ind the mean number of hours of cloud O cover per day#
Cambridge aper > F> H2.
*# he continuous random variable ' has probability density function given by
f ( x )={3
4( x2−1) ,∧1≤ x ≤2
0,∧otherwise(i) S"etch the probability density function of '#
(ii) Show that the mean& μ& of ' is #*?>.#(iii) Show that the standard deviation& & of ' is 2#--??& correct to ! decimal
places#
(iv)$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#
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?# he continuous random variable ' has probability density function given by
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
0,∧otherwisewhere " is a constant#
(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-
-# he continuous random variable ' has probability density function given by
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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
0,∧otherwisewhere k is a constant#
(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
,∧0≤ x ≤40,∧otherwise
where k is a constant#
(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
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