Download 25 Continuous Random Variables

25.
CONTINUOUS RANDOM VARIABLES
1.
The probability density function f(t) for the time, t hours, taken to complete a task can be
represented by the graph below:
f(t)
2/7
1
(a) [1 mark]
2
3
4
5
6
t
Calculate the probability that the task is completed in less than one hour.
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(b) [1 mark]
Calculate the probability that the task is completed in exactly 2 hours.
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(c) [1 mark]
What is the shortest time in which you can be certain the task is complete?
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(d)[2 marks] Calculate the probability that the task takes between 2 and 4 hours to
complete.
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(e) [3 marks] Given that the task is completed in less than 4 hours, what is the
probability of completing the task in less than 2 hours?
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25. Continuous Random Variables
2.
For each of the following functions, state, giving reasons, whether it could be the probability
density function of a continuous random variable.
 x
0  x  1

(a) [2 marks] f(x) = 2 - x 1 < x  2
 0
elsewhere

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x
 1  2 
 3  3 
 5
 x
5-x
(b) [2 marks] f(x) = 5Cx    
for x = 0, 1, 2 …
Note: 5Cx =  
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3.
The probability density function of a continuous random variable X is given by
a(x2  b) 0  x  2
f(x) = 
0
elsewhere

It is known that P(X < 1) =
,
where a and b are constants
1
, calculate
3
(a) [7 marks] a and b
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(b) [3 marks] P(X > 1.5)
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25. Continuous Random Variables
4.
(2, k)
f (x)
(4, k)
x
0
1
2
3
4
5
6
The probability density function for a random variable, X, is given by the above
graph.
(a) [2 marks] Find the value of k.
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(b) [1 marks] Find the probability that X is less than 2.
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(c) [2 marks] Given that X is less than 4, find the probability that X is less than 2.
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5.
The probability density function for a continuous random variable X is given by
 x ( x  2)( x  6)( x  7)


655.2
f ( x)  


0

0 x 6
.
elsewhere
(a) [2 marks] Find the following probabilities.
(i)
P(X < 2.6)
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(ii)
P(X > 3.2)
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(b) [3 marks] Is the distribution symmetric? Justify your answer.
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Vet Mathematics
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25. Continuous Random Variables
6.
For each of the following functions decide whether it can be a probability density function
of a continuous random variable. Justify your answer in each case.
2 e x
x0
 0
x0
(a) [2 marks] f ( x )  
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 e 3 3x

(b) [2 marks] f ( x)   x!

 0
x  {0,1,2,3,...}
otherwise
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 13
5 x  8
0 otherwise
(c) [2 marks] f ( x )  
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8  x
 30
(d) [2 marks] f ( x)  

 0
0  x  10
otherwise
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Vet Mathematics
194
25. Continuous Random Variables
7.
The time taken for an individual to respond to a particular stimulus is anywhere between
one and twenty seven seconds. The probability that an individual takes t seconds to
respond can be obtained from the following probability density function, f.
 (t  1) 2 (t 1)

f (t )   2 e

0
1  t  27
elsewhere
Calculate the probability that an individual responds in
(a) [2 marks] at most 15 seconds,
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(b) [2 marks] at least 7 seconds,
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(c) [2 marks] between 2.6 and 9 seconds.
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Vet Mathematics
195
25. Continuous Random Variables
8.
Lee never arrives at school before 8.00 am and never arrives after 9.00 am. The probability
distribution function for her time, t, of arrival at school is given below, with constant b and t
being measured as minutes after 8.00 am.
f(t)
b
b
f (t ) 
t
60
t
60
(a) What is the value of b ?
[2]
(b) Calculate the probability that Lee arrives at school
(i)
before 8.30 am,
[2]
(ii)
after 8.40 am.
[2]
(c) Given that Lee arrives at school after 8.30 am, what is the probability that she arrives
after 8.40 am?
[2]
(d) Find the time, T , to the nearest minute, such that the probability that Lee arrives at
school before T am is the same as the probability that she arrives at school after T
am.
[3]
Vet Mathematics
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25. Continuous Random Variables