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S2 – CONTINUOUS RANDOM VARIABLES – PAST EXAM QUESTIONS
1. The lifetime, X , in tens of hours, of a battery has a cumulative distribution function F( x ) given
by
F( )
2 x x x
- 5
x
x
x
(a) Find the median of X , giving your answer to 3 significant figures. (3)
(b) Find, in full, the probability density function of the random variable X. (3)
(c) Find P( X 1.2)
(2)
A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
(d) Find the probability that the lantern will still be working after 12 hours. (2) (Total 10 marks)
2. The random variable y has probability density function f( y ) given by
f ( )
kya y y otherwise
0 y 3
where k and a are positive constants.
(a) (i) Explain why a ≥ 3
(ii) Show that
a
k
(6)
Given that E( Y ) = 1.
(b) show that a = 4 and write down the value of k. (6)
For these values of a and k ,
(c) sketch the probability density function, (2)
(d) write down the mode of Y.
(1) (Total 15 marks)
3. A continuous random variable x has cumulative distribution function
F( x ) =
4
x
x
x x
(a) Find P( X < 0). (2)
(b) Find the probability density function f( x ) of X. (3)
(c) Write down the name of the distribution of X. (1)
(d) Find the mean and the variance of X. (3)
A random sample of size 3 is taken from the bag.
(b) List all the possible samples. (2)
(c) Find the sampling distribution of the mean value of the samples.
(6) (Total 11 marks)
6. The three independent random variables A , B and C each has a continuous uniform distribution
over the interval [0, 5].
(a) Find P( A > 3). (1)
(b) Find the probability that A , B and C are all greater than 3. (2)
The random variable Y represents the maximum value of A , B and C.
The cumulative distribution function of Y is
F(y) =
5
3
y
y
y y
(c) Find the probability density function of Y. (2)
(d) Sketch the probability density function of Y. (2)
(e) Write down the mode of Y. (1)
(f) Find E( Y ).
(3)
(g) Find P( Y > 3). (2) (Total 13 marks)
The diagram above shows a sketch of the probability density function f( x ) of the random
variable X. The part of the sketch from x = 0 to x = 4 consists of an isosceles triangle with
maximum at (2, 0.5).
(a) Write down E( X ). (1)
The probability density function f( x ) can be written in the following form.
f( x ) =
otherwise
x
x
b ax
ax
(b) Find the values of the constants a and b. (2)
(c) Show that σ, the standard deviation of X , is 0.816 to 3 decimal places. (7)
9. A random variable X has probability density function given by
0 otherwise
f ( )
x x x
(a) Show that the cumulative distribution function F( x ) can be written in the form
ax
2
- bx + c , for 1 x 4 where a , b and c are constants. (3)
(b) Define fully the cumulative distribution function F( x ).
(2)
(c) Show that the upper quartile of X is 2.5 and find the lower quartile. (6)
Given that the median of X is 1.
(d) describe the skewness of the distribution. Give a reason for your answer. (2) (Total 13 marks)
10. A random variable X has probability density function given by
0 otherwise
f ( )
3 kx x
x x
x
where k is a constant.
(a) Show that 5
k
(4)
(b) Calculate the mean of X. (4)
(c) Specify fully the cumulative distribution function F( x ).
(7)
(d) Find the median of X. (3)
(e) Comment on the skewness of the distribution of X. (2) (Total 20 marks)
11. The continuous random variable Y has cumulative distribution function F( y ) given by
F )
4 2
y
k y y y
y
(y
(a) Show that 18
k .
(2)
(b) Find P( Y > 1.5).
(2)
(c) Specify fully the probability density function f( y ). (3) (Total 7 marks)
12. The continuous random variable X has probability density function f( x ) given by
0 otherwise
f ( )
x x x
(a) Sketch f( x ) for all values of x. (3)
(d) Using your answer to part (c), find the median of X.
(3) (Total 14 marks)
14. The continuous random variable X is uniformly distributed over the interval α < x < β.
(a) Write down the probability density function of X , for all x. (2)
(b) Given that E( X ) = 2 and P( X < 3) = 8
find the value of α and the value of β.
(4)
A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one
end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable X. Find
(c) E( X ), (1)
(d) the standard deviation of X , (2)
(e) the probability that the shorter piece of wire is at most 30 cm long. (3) (Total 12 marks)
15. The continuous random variable X has cumulative distribution function
F(x)
2 3
x
x
x
x x
(a) Find P( X > 0.3).
(2)
(b) Verify that the median value of X lies between x = 0.59 and x = 0.60. (3)
(c) Find the probability density function f( x ). (2)
(d) Evaluate E( X ).
(3)
(e) Find the mode of X. (2)
(f) Comment on the skewness of X. Justify your answer. (2) (Total 14 marks)
16. The continuous random variable X has probability density function
otherwise.
f
x
k
x
x
(a) Show that k =. 2
(3)
(b) Specify fully the cumulative distribution function of X. (5)
(c) Calculate E( X ). (3)
18. A continuous random variable X has probability density function f( x ) where
f( x ) =
0 , otherwise,
3
k x x x
where k is a positive constant.
(a) Show that k =
(4)
Find
(b) E( X ), (3)
(c) the mode of X ,
(3)
(d) the median of X. (4)
(e) Comment on the skewness of the distribution. (2)
(f) Sketch f( x ).
(2) (Total 18 marks)
19. The random variable X has probability density function
f( x ) =
0 , otherwise.
2 k x x x
(a) Show that k = 9
2 .
(3)
Find
(b) E( X ), (3)
(c) the mode of X.
(2)
(d) the cumulative distribution function F( x ) for all x. (5)
(e) Evaluate P( X 2.5). (2)
(f) Deduce the value of the median and comment on the shape of the distribution. (2) (Total 17 marks)
20. A random variable X has probability density function given by
f( x ) =
0 , otherwise.
3
x
x
x
(a) Calculate the mean of X. (5)
(b) Specify fully the cumulative distribution function F( x ).
(7)
(c) Find the median of X. (3)
22. A continuous random variable X has probability density function f( x ) where
f( x ) =
0 , otherwise
2
k x x x
where k is a positive integer.
(a) Show that k = 3. (4)
Find
(b) E( X ), (4)
(c) the cumulative distribution function F( x ),
(4)
(d) P(0.3 < X < 0.3). (3) (Total 15 marks)
23. The continuous random variable X has cumulative distribution function
F( )
2 2 3
1 x x x
x
x
x
(a) Find P( X > 0.7).
(2)
(b) Find the probability density function f( x ) of X. (2)
(c) Calculate E( X ) and show that, to 3 decimal places, Var ( X ) = 0.057. (6)
One measure of skewness is
Standard deviation
Mean Mode
(d) Evaluate the skewness of the distribution of X. (4) (Total 14 marks)
24. The lifetime, in tens of hours, of a certain delicate electrical component can be modelled by the
random variable X with probability density function
0 , otherwise.
f () 7
1
42
1
x
x x
x
(median =) 1.26 A1 3
Note
M1 putting F( x ) = 0.
M1 using correct quadratic formula. If use calc need to get 1.26 (384...) A1 cao 1.26 must reject the other root.
If they use Trial and improvement they have to get the correct answer
to gain the second M mark.
(b) Differentiating
d 2 3
d 9
x x
x
x
^
^ M1 A
f ( ) 9
0 otherwise
x x
x
^ ^
B1ft 3
Note
M1 attempt to differentiate. At least one x
n → x
n – 1
A1 correct differentiation
B1 must have both parts- follow through their F′( x ) Condone <
(c) P( X ≥ 1.2) = 1 – F(1.2) M
= 1 – 0.
, 0.6267 awrt 0.627 A1 2
Note
M1 finding/writing 1 – F(1.2) may use/write
5
2
1 d 9
x x
or 1 – ^
- 2
1
1 d 9
x x or
5
2
"^ theirf( x )"d x .Condone missing d x
A1 awrt 0.
(d) (0.6267)
4 = 0.154 awrt 0.154 or 0.155 M1 A1 2
Note
M1 (c)
4 If expressions are not given you need to check the calculation
is correct to 2sf. A1 awrt 0.154 or 0. [10]
2. (a) (i) f( y ) > 0 or f(3) > 0 M
ky a y > 0 or 3 k ( a – 3) > 0 or ( a – y ) > 0 or ( a – 3) > 0
a > 3 A1 cso
Note
M1 for putting f( y ) ≥ 0 or f(3) ≥ 0 or ky ( a – y ) ≥ 0 or
3 k ( a – 3) ≥ 0 or ( a – y ) ≥ 0 or ( a – 3) ≥ 0 or state in words the probability can not be negative o.e.
A1 need one of ky ( a – y ) ≥ 0 or 3 k ( a – 3) ≥ 0 or ( a – y ) ≥ 0
or ( a – 3) ≥ 0 and a ≥ 3
(ii)
3 2
0
k ay ( y ) dy 1
integration M
3 2 3
0
ay y
k
^
^
answer correct A
a
k
^
answer = 1 M
k
^ a^
k
a
Note
M1 attempting to integrate (at least one y
n → y
n + 1 )
(ignore limits) A1 Correct integration. Limits not needed. And equals
1 not needed.
M1 dependent on the previous M being awarded. Putting equal to 1 and have the correct limits. Limits do not need to
be substituted.
A1 cso
