Reference no: EM132710412
QUESTION ONE
a) If X is the sum of the scores of two dice, the sample space for X will be;
X 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
Find the probability distribution of X
(b)i)
|
X
|
1
|
2
|
3
|
4
|
5
|
|
P(X=x)
|
0.2
|
0.1
|
0.3
|
a
|
0.1
|
Find a and P (X≥2)
ii)
|
X
|
-2
|
-1
|
0
|
1
|
|
P(X=x)
|
k
|
2k
|
2k
|
k
|
Find k and p(X≤0)
c)The random variable, X has probability distribution
p{X=k} =|(k-1)/36 k=2,3,4,5,6,7
p(X=k)= |(13-k)/36 k=8,9,10,11,12
Find a)P(X=3) b)P(X)>9
QUESTION TWO
a) The random variable X ,has cumulative distribution function given by the table
|
X
|
-3
|
-2
|
-1
|
0
|
1
|
2
|
3
|
|
F(x)
|
0.2
|
0.3
|
0.5
|
0.6
|
0.75
|
0.9
|
1
|
Find P(X=2) and P(|X|<2)
b) The random variable ,X, has cumulative distribution function in the form
F(x) = (x+a)2/25 x=1,2,3, a>0
Find the value of a, and give the probability distribution of X
c) For the following cumulative distribution functions , write down the probability distribution
|
X
|
-2
|
-1
|
0
|
1
|
2
|
|
F(x)
|
0.1
|
0.2
|
0.4
|
0.7
|
1
|
|
X
|
5
|
7
|
10
|
15
|
20
|
|
F(x)
|
1/4
|
1/3
|
1/2
|
3/4
|
1
|
QUESTION THREE
a) X is a discrete random variable with probability distribution
Show that k=0.1 and calculate E[X] and E[X2]
b) X is a probability distribution function and E[X]=5.7
|
X
|
1
|
2
|
4
|
8
|
16
|
|
P(X=x)
|
0.1
|
a
|
0.3
|
b
|
0.1
|
Find a ,b and Var [X]
c) The random variable ,X, has the probability function
P(X=x)=(11-2x)/2 x=1,2,3,4,5
i) Construct a table giving the probability distribution of X.
Find;
ii) P(2<X<5)
iii) E[X]
iv) Var[X]
iv) Var[2-3X]
QUESTION FOUR
A supermarket sells top up vouchers valued at £5 £10 £15 or £20 .The value ,in pounds of a top up voucher sold may be regarded as a random variable ,X, with the following probability distribution;
|
X
|
5
|
10
|
15
|
20
|
|
P(X=x)
|
0.20
|
0.40
|
0.15
|
0.25
|
a) Find the mean and standard deviation of X
b) What is the probability that the next top up voucher sold is less than £15
c) As a promotion the marketing department decides the supermarket will offer 10% off the cost of each top up voucher. Write down the mean and the standard deviation of the value of vouchers sold in the promotion, assuming customers continue to use the same buying pattern. The marketing department had originally planned to offer £ I off each voucher.
d) i)What would have been the mean and standard deviation of the value of the vouchers sold ,assuming customers continued to use the same buying pattern.
ii) An experienced manager had sold the supermarket that they would only sell £5 top up vouchers if they used this strategy. Explain why he thought this would happen.