Download probability density functions

Created by T Madas
PROBABILITY
DENSITY
FUNCTIONS
Created by T. Madas
Created by T Madas
P.D.F.
CALCULATIONS
Created by T. Madas
Created by T Madas
Question 1 (***)
The lifetime of a certain brand of battery, in tens of hours, is modelled by the
continuous random variable X with probability density function f ( x ) given by
 2
 75 x

f ( x) =  2
 15

0
0≤ x≤5
5 < x ≤ 10
otherwise
a) Sketch f ( x ) for all x .
b) Determine P( X > 4) .
Two such batteries are needed by a piece of electronic equipment. This equipment will
only operate if both batteries are still functional.
c) If two new batteries are fitted to this equipment, determine the probability that
this equipment will stop working within the next 40 hours.
FS1-D , P( X > 4) =
Created by T. Madas
59 2144
,
≈ 0.381
75 5625
Created by T Madas
Question 2 (***)
The lengths of telephone conversations, in minutes, by sales reps of a certain company
are modelled by the continuous random variable T .
The probability density function of T is denoted by f ( t ) , and is given by
 kt
f (t ) = 
0
a) Show that k =
0 ≤ t ≤ 12
otherwise
1
.
72
b) Determine P(T > 5) .
c) Show by calculation that E (T ) = Var (T ) .
d) Sketch f ( t ) for all t .
A statistician suggests that the probability density function f ( t ) as defined above,
might not provide a good model for T .
e) Give a reason for his suggestion.
FS1-A , P(T > 5) =
Created by T. Madas
119
, E (T ) = Var (T ) = 8
144
Created by T Madas
Question 3 (****)
The continuous random variable X has probability density function f ( x ) , given by
 2x + k
f ( x) = 
0

3≤ x ≤ 4
otherwise
a) Show that k = −6 .
b) Sketch f ( x ) for all x .
c) State the mode of X .
d) Calculate, showing detailed workings, the value of …
i. … E ( X ) .
ii. … Var ( X ) .
iii. … the median of X .
e) Determine with justification the skewness of the distribution.
2
mode = 4 , E ( X ) = 11 ≈ 3.67 , Var ( X ) = 1 ≈ 0.0556 , median = 3 +
≈ 3.71 ,
3
18
2
mean < median < mode ⇒ negative skew
Created by T. Madas
Created by T Madas
Question 4 (****+)
The continuous random variable X has probability density function f ( x ) , given by
 kx 16 − x 2
f ( x) = 

0
(
a) Show that k =
)
0≤ x≤4
otherwise
1
.
64
b) Calculate, showing detailed workings, the value of …
i. … E ( X ) .
ii. … Var ( X ) .
c) Show by calculation, that the median is 2.165 , correct to 3 decimal places.
d) Use calculus to find the mode of X .
e) Sketch the graph of f ( x ) for all x .
f) Determine with justification the skewness of the distribution.
E ( X ) = 32 ≈ 2.13 , Var ( X ) = 176 ≈ 0.782 , mode ≈ 2.31 ,
15
225
mean < median < mode ⇒ negative skew
Created by T. Madas
Created by T Madas
Question 5 (****+)
The continuous random variable X has probability density function f ( x ) , given by
 kx ( a − x )
f ( x) = 
0

0≤ x≤4
otherwise
where k and a are positive constants.
A statistician claims that a ≥ 4 .
a) Justify the statistician’s claim.
b) Show clearly that
k=
3
.
8 ( 3a − 8 )
k=
9
.
80 ( a − 3)
It is further given that E ( X ) = 2.4 .
c) Show further that
d) Hence determine the value of a and the value of k .
e) Sketch the graph of f ( x ) for all x .
f) Find the mode of X .
a = 6 , k = 3 = 0.0375 , mode = 3
80
Created by T. Madas
Created by T Madas
Question 6 (****+)
The continuous random variable X has probability density function f ( x ) , given by
 2 x
 21
f ( x) =  2 (6 − x)
 15
 0
0≤ x≤k
k <x≤6
otherwise
Determine the value of the positive constant k , and hence or otherwise find
P  X < 13 k X < k  .
FS1-U , k = 3.5 , P  X < 1 k X < k  = 1
3

 9
Created by T. Madas
Created by T Madas
Question 7 (****+)
f ( x)
P
4
O
Q ( a,0 )
x
The figure above shows the graph of the probability density function f ( x ) of a
continuous random variable X .
The graph consists of the curved segment OP with equation
f ( x ) = kx 2 , 0 ≤ x ≤ 4 ,
where k is a positive constant.
The graph of f ( x ) is a straight line segment from P to Q ( a,0 ) , for 4 < x ≤ a , where
a is a positive constant.
For all other values of x , f ( x ) = 0 .
a) State the mode of X .
It is given that the mode of X is equal to the median of X .
b) Show that k =
3
, and find the value of a .
128
[continues overleaf]
Created by T. Madas
Created by T Madas
[continued from overleaf]
It is further given that E ( X ) =
71
.
18
c) Determine with justification the skewness of the distribution.
mode = 4 , a = 20 , mean < median = mode ⇒ negative skew
3
Created by T. Madas
Created by T Madas
Question 8 (****+)
y
O
3
6
x
The figure above shows the graph of a probability density function f ( x ) of a
continuous random variable X .
The graph consists of two straight line segments of equal length joined up at the point
where x = 3 .
The probability density function f ( x ) is fully specified as
 ax

f ( x ) =  b + cx
0

0≤ x≤3
3< x ≤6
otherwise
where a , b and c are non zero constants.
a) Show that b = 2 , c = − 1 and find the value of a .
3
9
b) State the value of E ( X ) .
c) Show that Var ( X ) = 1.5 .
[continues overleaf]
Created by T. Madas
Created by T Madas
[continued from overleaf]
d) Determine the upper and lower quartile of X .
A statistician claims that P ( X − µ < σ ) > 0.5 .
e) Show that the statistician’s claim is correct.
FS1-W , a = 1 , E ( X ) = 3 , Q1 ≈ 2.121 , Q3 ≈ 3.879
9
Created by T. Madas
Created by T Madas
Question 9 (****+)
The continuous random variable X has probability density function f ( x ) , given by
 kx
f ( x) = 
0
0≤x≤a
otherwise
where a and k are positive constants.
Show, by a detailed method, that
Var ( X ) = 1 a 2 .
18
FS1-Z , proof
Created by T. Madas
Created by T Madas
P.D.F. to C.D.F
CALCULATIONS
Created by T. Madas
Created by T Madas
Question 1 (****)
The continuous random variable X has probability density function f ( x ) , given by
 2 ( 5 − x ) 2 ≤ x ≤ 5
f ( x) = 9
otherwise
 0
The cumulative distribution function of X , is denoted by F ( x ) .
a) Find and specify fully F ( x ) .
b) Use F ( x ) , to show that the lower quartile of X is approximately 2.40 , and
find the value of the upper quartile.
c) Given that the median of X is 2.88 , comment on the skewness of X .
0

 1
FS1-B , F ( x ) = − ( x − 8 )( x − 2 )
9

1

x<2
2 ≤ x ≤ 5 , Q3 = 3.5 ,
x>5
Q2 − Q1 < Q3 − Q2 ⇒ +ve skew
Created by T. Madas
Created by T Madas
Question 2 (****)
The continuous random variable X has probability density function f ( x ) , given by
kx ( x − 3) 3 ≤ x ≤ 4
f ( x) = 
0
otherwise

a) Use integration to show that k = 6 .
11
b) Calculate the value of E ( X ) .
c) Show that Var ( X ) = 0.053 , correct to three decimal places.
The cumulative distribution function of X , is denoted by F ( x ) .
d) Find and specify fully F ( x ) .
e) Show that the median of X lies between 3.70 and 3.75 .
0



FS1-H , E ( X ) = 81 ≈ 3.68 , F ( x ) =  1 2 x3 − 9 x 2 + 27
22
11

1

(
Created by T. Madas
x<3
)
3≤ x ≤ 4
x>4
Created by T Madas
Question 3 (****)
The continuous random variable X has probability density function f ( x ) , given by
 k x + 2 )2
f ( x) =  (
0

−2≤ x ≤0
otherwise
a) Show clearly that k = 3 .
8
b) Find the value of E ( X ) .
c) Show that Var ( X ) = 0.15 .
The cumulative distribution function of X , is denoted by F ( x ) .
d) Find and specify fully F ( x ) .
e) Determine P ( −1 ≤ X ≤ 1) .
0

1 3 3 2 3
E ( X ) = −0.5 , F ( x ) =  x + x + x + 1
8
4
2

1

x < −2
− 2 ≤ x ≤ 0 , P ( −1 ≤ X ≤ 1) = 0.875
x>0
Created by T. Madas
Created by T Madas
Question 4 (****)
The continuous random variable X has probability density function f ( x ) , given by:
1 3
 x
f ( x ) =  60
 0
2≤ x≤4
otherwise
a) Find the value of E ( X ) .
b) Show that the standard deviation of X is 0.516 , correct to 3 decimal places.
The cumulative distribution function of X , is denoted by F ( x ) .
c) Find and specify fully F ( x ) .
d) Determine P ( X > 3.5) .
e) Calculate the median of X , correct to two decimal places.
 0

E ( X ) = 248 ≈ 3.31 , F ( x ) =  1 x 4 − 16
75
240

 1
(
x<2
)
2 ≤ x ≤ 4 , P ( X > 3.5 ) = 113 ,
256
x>4
median ≈ 3.41
Created by T. Madas
Created by T Madas
Question 5 (****)
The continuous random variable X has probability density function f ( x ) , given by
 kx ( 6 − x )
f ( x) = 
0

a) Show clearly that k =
0≤ x≤5
otherwise
3
.
100
b) Calculate the value of E ( X ) .
c) Find the mode of X .
The cumulative distribution function of X , is denoted by F ( x ) .
d) Find and specify fully F ( x ) .
e) Verify that the median of X lies between 2.85 and 2.9 .
f) Determine with justification the skewness of the distribution.
 0

E ( X ) = 45 , mode = 3 , F ( x ) =  1 x 2 ( 9 − x )
100
16

 1
x<0
0≤ x≤5 ,
x>5
mean < median < mode ⇒ negative skew
Created by T. Madas
Created by T Madas
Question 6 (****)
The continuous random variable X has probability density function f ( x ) , given by
 k ( x − 1)( x − 4 )
f ( x) = 
0

1≤ x ≤ 4
otherwise
a) Show clearly that k = − 2
9
b) Sketch the graph of f ( x ) , for all x .
c) State the value of E ( X ) .
d) Calculate the Var ( X ) .
The cumulative distribution function of X , is denoted by F ( x ) .
e) Find and specify fully F ( x ) .
f) Determine with justification the skewness of the distribution.
E ( X ) = 2.5 , Var ( X ) = 0.45
0

 1
F ( x) = 
11 − 24 x + 15 x 2 − 2 x3
27

1

(
x <1
)
1≤ x ≤ 4 ,
x>4
mean = median = mode = 2.5 ⇒ zero skew
Created by T. Madas
Created by T Madas
Question 7 (****)
The continuous random variable X has probability density function f ( x ) , given by
 2+ x

f ( x) =  k
 0
2≤ x≤5
otherwise
a) Show clearly that k = 33 .
2
b) Find the value of E ( X ) .
c) Show that Var ( X ) = 0.731 , correct to three decimal places.
The cumulative distribution function of X , is denoted by F ( x ) .
d) Find and specify fully F ( x ) .
e) Determine the median of X .
0


E ( X ) = 40 , F ( x ) =  1 ( x + 6 )( x − 2 )
33
11

1

Created by T. Madas
x<2
2 ≤ x ≤ 5 , median ≈ 3.70
x>5
Created by T Madas
Question 8 (****)
The continuous random variable X has probability density function f ( x ) , given by
 1
 3

f ( x ) =  4 x3
195

 0

0≤ x≤2
2< x≤3
otherwise
a) Show that E ( X ) = 1.532 , correct to three decimal places.
The cumulative distribution function of X , is denoted by F ( x ) .
b) Find and specify fully F ( x ) .
c) Calculate the median of X .
d) Determine with justification the skewness of the distribution.
0


1x

3
FS1-E , F ( x ) = 
4
1
195 x + 38
65

1

x<0
0≤ x≤2
, median = 1.5 ,
2< x≤3
x>3
(mode) < median < mean ⇒ slight positive skew
Created by T. Madas
Created by T Madas
Question 9 (****)
The continuous random variable X has probability density function f ( x ) , given by
1 x
 4
f ( x ) =  kx3

0
0≤ x≤2
2<x≤4
otherwise
a) Show clearly that k = 1 .
120
The cumulative distribution function of X , is denoted by F ( x ) .
b) Find and specify fully F ( x ) .
c) State the median of X .
d) Show that E ( X ) = 58 .
25
e) Calculate the interquartile range of X .
0

 1 2
 8x

F ( x) =  4
x + 224

 480
 1
x<0
0≤ x≤2
, median = 2 , IQR ≈ 2.00
2< x≤4
x>4
Created by T. Madas
Created by T Madas
Question 10 (****)
The continuous random variable X has probability density function f ( x ) , given by
 k x2 − 2 x + 3


f ( x) = 
1k
3


0
(
)
0≤ x<2
2≤ x≤3
otherwise
a) Show clearly that k = 1 .
5
The cumulative distribution function of X , is denoted by F ( x ) .
b) Find and specify fully F ( x ) .
c) Show that E ( X ) = 11 .
10
d) Show that the median of X lies between 1.05 and 1.1.
0

1
2
15 x x − 3 x + 9
F ( x) = 
1 ( x + 12 )

15

1

(
Created by T. Madas
x<0
)
0≤ x<2
2≤ x≤3
x>4
Created by T Madas
Question 11 (****+)
The continuous random variable X has probability density function f ( x ) , given by:
 1 x
 10
f ( x) = 2 − 2 x
5

 0
0≤x<4
4≤ x≤5
otherwise
a) Sketch the graph of f ( x ) for all x .
b) State the mode of X .
c) Show clearly that E ( X ) = 3 .
d) Calculate the value of Var ( X ) .
e) Find and specify fully the cumulative distribution function of X , F ( x ) .
0


1 x2

20
mode = 4 , Var ( X ) = 7 , F ( x ) = 
6
2
− 15 x − 10 x + 20

1

(
Created by T. Madas
x<0
0≤ x<4
)
4≤ x≤5
x>5
Created by T Madas
Question 12 (****+)
The continuous random variable X has probability density function f ( x ) , defined by
the piecewise continuous function
 1 ( x − 1)
12
1

f ( x) =  6
5 − 1 x
12 24
0
1≤ x ≤ 3
3< x ≤6
6 < x ≤ 10
otherwise
a) Sketch the graph of f ( x ) , for all values of x .
The cumulative distribution function of X , is denoted by F ( x ) .
b) Show by detailed calculations that
0

 1 2 1
1
 24 x − 12 x + 24

F ( x) =  1 x − 1
6
3

 − 1 x 2 + 5 x − 13
12
12
 48
1

x <1
1≤ x ≤ 3
3< x ≤6
6 < x ≤ 10
x > 10
[continues overleaf]
Created by T. Madas
Created by T Madas
[continued from overleaf]
c) Calculate the median of X .
d) Determine the value of P ( 2 < X < 4 ) ∪ P ( 5 < X < 9 ) .
e) Find the value for E ( X ) .
FS1-V , median = 5 , P ( 2 < X < 4 ) ∪ P ( 5 < X < 9 ) = 37 , E ( X ) = 61
48
12
Created by T. Madas
Created by T Madas
C.D.F. to P.D.F
CALCULATIONS
Created by T. Madas
Created by T Madas
Question 1 (***)
The cumulative distribution function F ( x ) of a continuous random variable X is
given by the following expression
0


F ( x ) = k x 4 + 2 x 2 − 3

1

(
x <1
)
1≤ x ≤ 3
x>3
where k is a positive constant.
a) Show clearly that k =
1
.
96
b) Find P ( X > 2 ) .
c) Determine the probability density function of X , for all values of x .
1 3
 24 x + x
25
P( X > 2) =
, f ( x) = 
32
0

(
Created by T. Madas
)
1≤ x ≤ 3
otherwise
Created by T Madas
Question 2 (***)
The cumulative distribution function F ( x ) of a continuous random variable X is
given by
0


F ( x ) = k ( x + 3)

1

x < −3
−3≤ x ≤ 8
x >8
where k is a positive constant.
a) Show clearly that k = 1 .
11
b) Calculate P ( X < 1) .
c) Find P ( X = 1) .
d) Define the probability density function of X , for all values of x .
e) State the name of this probability distribution.
f) Determine the value of E ( X ) and the value of Var ( X ) .
 1
P( X < 1) = 4 , P( X = 1) = 0 , f ( x ) = 11
11
 0
−3 ≤ x ≤ 8
, uniform continuous ,
otherwise
E ( X ) = 5 , Var ( X ) = 121
2
12
Created by T. Madas
Created by T Madas
Question 3 (***)
The continuous random variable T represents the lifetime, in tens of hours, for a
certain brand of battery.
The cumulative distribution function F ( t ) of the variable T is given by
0

1
F ( t ) =  ( t + 6 )( t − 2 )
20

1

t<2
2≤t ≤4
t>4
a) Calculate …
i. … P (T > 3.5 ) .
ii. … P (T = 3) .
b) Show that the median of T is 3.10 , correct to three significant figures.
c) Define the probability density function of T , for all values of t .
Four new batteries whose lifetime is modelled by T are fitted to a road hazard lantern.
This lantern will only remain operational if all four batteries are working.
d) Determine the probability that the lantern will operate for more than 35 hours.
 1 ( t + 2 )
P (T > 3.5 ) = 23 , P(T = 3) = 0 , f ( t ) = 10
80
 0
Created by T. Madas
2≤t ≤4
otherwise
, 0.00683
Created by T Madas
Question 4 (***)
The cumulative distribution function, F ( y ) , of a continuous random variable Y is
given by
 0

F ( y ) =  1 y3
64

 1
y<0
0≤ y≤4
y>4
a) Find the probability density function of Y for all y .
b) Calculate the value of E (Y ) .
c) Show that Var (Y ) = 3 .
5
d) State the mode of Y .
 3 y 2
f ( y ) =  64
 0
0≤ y≤4
otherwise
Created by T. Madas
, E (Y ) = 3 , mode = 4
Created by T Madas
Question 5 (***+)
The time, in weeks, a patient has to wait for an appointment to see a psychiatrist in a
certain hospital is modelled by the continuous random variable T .
The cumulative distribution function of T is given by
 0

F (t ) =  1 t3
1728

 1
t<0
0 ≤ t ≤ 12
t > 12
a) Find, to the nearest day, the time within which 80% of the patients have been
given an appointment.
b) Define the probability density function of T , for all values of t .
c) Calculate the mean and the variance of T .
d) Determine the probability that the time a patient has to wait for an appointment,
is more than one standard deviation above the mean.
 1 t 2
≈ 78 days , f ( t ) =  576
 0
0 ≤ t ≤ 12
, E (T ) = 9, Var (T ) = 5.4 , ≈ 0.1597
otherwise
Created by T. Madas
Created by T Madas
Question 6 (***+)
The time, in hours, spent on a piece of homework by a group of students is modelled
by the continuous random variable T .
The cumulative distribution function of T is given by
0


F ( t ) =  k 2t 3 − t 4

1

(
t<0
)
0 ≤ t ≤ 1.5
t > 1.5
where k is a positive constant.
a) Show clearly that k = 16 .
27
b) Find the proportion of students who spent over an hour in their homework.
c) Verify that the median is 0.921 , correct to 3 decimal places.
d) Define the probability density function of T , for all values of t .
e) Show that E (T ) = 0.9 .
f) Determine with justification the skewness of the distribution.
 32 3t 2 − 2t 3
P(T > 1) = 11 , f ( t ) =  27
27
0

(
)
0 ≤ t ≤ 1.5
otherwise
mean < median < (mode) ⇒ negative skew
Created by T. Madas
,
Created by T Madas
Question 7 (***+)
A continuous random variable X has the following cumulative distribution function
F ( x ) , defined by
0

 1 2
F ( x ) =  x 6 − x2
5

1

(
x<0
)
0 ≤ x ≤1
x >1
a) Find P ( X > 0.5 ) .
b) Define f ( x ) , the probability density function of X , for all values of x .
c) Calculate the mean of X .
d) Show that the standard deviation of X is 0.239 , correct to 3 decimal places.
A skewness coefficient can be calculated by the formula
mean − mode
.
standard deviation
e) Evaluate the skewness coefficient for this distribution.
P ( X > 0.5) = 57
80
 4 x 3 − x 2
f ( x) =  5
0

(
)
0 ≤ x ≤1
otherwise
Created by T. Madas
, mean = 0.64 , −1.507
Created by T Madas
Question 8 (****)
The cumulative distribution function, F ( x ) , of a continuous random variable X is
given by the following expression
0


F ( x ) =  k − x3 + 6 x 2 − 5

1

(
x <1
)
1≤ x ≤ 4
x>4
where k is a positive constant.
a) Show clearly that k =
1
.
27
b) Define f ( x ) , the probability density function of X , for all values of x .
c) Show that the mean of X is 2.25 .
d) Sketch the graph of f ( x ) for all x , and hence state the mode of X .
e) Show by calculation that the median of X lies between its mode and its mean.
f) State the skewness of the distribution.
 1 x ( 4 − x )
f ( x) =  9
0

1≤ x ≤ 4
otherwise
Created by T. Madas
, mode = 2 , positive skew
Created by T Madas
Question 9 (****)
The cumulative distribution function F ( x ) of a continuous random variable X is
given by the following expression
 0

F ( x ) =  kx ( x + 2 )
 1

x<0
0≤ x≤2
x>2
where k is a positive constant.
a) Find the value of k .
b) Show that the median of X is 1.236 , correct to 3 decimal places.
c) Define the probability density function of X , f ( x ) , for all x .
d) Show that the mean of X is 7 .
6
e) Calculate the variance of X .
f) Sketch the graph of f ( x ) for all x , and hence state the mode of X .
 1 ( x + 1)
1
k = , f ( x) =  4
8
 0
0≤ x≤2
otherwise
Created by T. Madas
, Var ( X ) = 11 , mode = 2
36
Created by T Madas
Question 10 (****)
The cumulative distribution function F ( x ) of a continuous random variable X is
given by the following expression
0


F ( x ) =  ( 3 − x )( x − 1)2

1

x <1
1≤ x ≤ 2
x>2
a) Find P ( X > 1.2 ) .
b) Show that the median of X lies between 1.59 and 1.60 .
c) Show that for 1 ≤ x ≤ 2 , the probability density function of X , f ( x ) , is
f ( x ) = (1 − x )( 3x − 7 ) .
d) Show further that the mean of X is
e) Given that the variance of X is
19
.
12
43
, find the exact value of E X 2 .
720
( )
f) Find the mode of X .
( )
P ( X > 1.2 ) = 0.928 , E X 2 = 77 , mode = 5
30
3
Created by T. Madas
Created by T Madas
Question 11
(*****)
The continuous random variable X has the following cumulative distribution function
 0

F ( x ) = ax − bx 2
 1

x<0
0≤ x≤k
x>k
where vehicles a , b and k are positive constants.
The variable Y is related to X by
Y = 3X − 2 .
Determine the value of a , b and k given further that E (Y ) = 2 and Var (Y ) = 8 .
FS1-T , a = 1 , b = 1 , k = 4
2
16
Created by T. Madas