Determine the value of c for the distribution of random variable X that has a discrete distribution...

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Determine the value of c for the distribution of random variable X that has a discrete distribution with the following probability function, f(x) =  cx for x = 1, . . . , 5 0 otherwise [2 marks] Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley XXXXXXXXXXb) Let X be the ticket price for an event. supose that the probability distribution of X is: X XXXXXXXXXX 200 p XXXXXXXXXX XXXXXXXXXXi) What is the probability that a randomly selected attendee paid more that $140 for the ticket? [2 marks] (ii) What is the probability that a randomly selected attendee paid less than $160? [2 marks] (iii) Compute the expected value and standard deviation of X. [4 marks] Data Source: Statistics and data with R. Y. Cohen & J. Cohen, Wiley XXXXXXXXXXQuestion 2 - 6 marks Suppose that a box contains 7 red balls and and 3 blue balls. If 5 balls are selected at random, without replacement, determine the probability function of the number of red balls that will be obtained using (a) Counting methods 3 marks (b) Simulation 3 marks . Data Source: Probability and Statistics M. DeGroot & M. Schervish, Addison-Wesley XXXXXXXXXXQuestion 3 - 8 marks Suppose that the life expectancy X of each member of a group of people is a random variable having an exponential distribution with parameter ? = 1 50 years. (a) For an individual from this group, compute the probability that: (i) He will survive to 65, 2 marks (ii) He will live to be at least 70 years old, given that he just celebrated his 40th birthday, 3 marks (b) For what value of c is P(X > c) = 1 2 ? 3 marks f(x|?) = ?e-?x ? > 0, x > 0 Data Source: A Course in Mathematical Statistics. G Roussas, Academic Press XXXXXXXXXXQuestion 4 - 6 marks A uniform distribution is defined with density function f(x; a, ß) = 1 (ß - a) ß > a and x ? (a, ß) Show that (a) E(X) = a + ß 2 [2 marks] (b) var(X) = (ß - a) 2 12 [4 marks] Data Source: A Course in Mathematical Statistics. G Roussas, Academic Press XXXXXXXXXXQuestion 5 - 6 marks Let X be a random variable with probability density function f(x) = ?e-?(x-a) x > a (a) Find its Moment Generating Function MX(t) for those t’s that exist. [3 marks] (b) Calculate E(X) [1 mark] (c) Calculate s 2 (X) [2 marks] Data Source: A Course in Mathematical Statistics. G Roussas, Academic Press (1997)

David · Accepted Answer

Q1 (a)
	Determine the value of c for the distribution of random variable X that has a discrete
distribution with the following probability function,
f(x) = 
1,2,3,4,5
0
cxforx
Otherwise
=
ì
í
î
Solution: The probability distribution table:
X
1
2
3
4
5
P(X)
c
2c
3c
4c
5c
 We know that ∑P(x) = 1
                        c+2c+3c+4c+5c = 1
                         15c = 1
 Answer: c = 1/15
	
	(i) What is the probability that a randomly selected attendee paid more that $140 for the ticket?
P(x>140) = P(x = 160)+ P(x = 180) + P(x = 200)
                =0.16 + 0.13 + 0.11
                = 0.40
Answer: P(x>140) = 0.40
	
	(ii) What is the probability that a randomly selected attendee paid less than $160?
P(x
³
æö
=
ç÷
>>
èø
                 = 
70/50
40/50
e
e
-
-
                 =e(-7/5)+(4/5)
                = e-3/5   
                = e-0.6   
               = 0.549
	3(b)
	(b) P(x>c)= ½
    
1
2
x
c
edx
l
l
¥
-
=
ò
 
    
0.7
c
ee
l
--
=
   - 
l
c = -0.7
   Answer:    c = 
0.7
l
 if 
l
 = 
1
50
 then c = 35
	4.
	A uniform distribution is defined with density function
(
)
1
fx
ba
=
-
     
b
> x >
a
 
Its Moment generating function of variate x  is given by
Mx(t) = E(etx)
         = 
()
tx
efxdx
b
a
ò
         = 
1
.
tx
edx
b
a
ba
-
ò
        Mx(t) = 
(
)
11
..

Determine the value of c for the distribution of random variable X that has a discrete distribution with the following probability function, f(x) = cx for x = 1, . . . , 5 0 otherwise [2 marks] Data...

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Determine the value of c for the distribution of random variable X that has a discrete distribution with the following probability function, f(x) =  cx for x = 1, . . . , 5 0 otherwise [2 marks] Data...

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Determine the value of c for the distribution of random variable X that has a discrete distribution with the following probability function, f(x) = cx for x = 1, . . . , 5 0 otherwise [2 marks] Data...