Answer:
- four points means 4 values
- chosen uniformly means chosen consecutively or one after the other, without skipping any value
- chosen independently means that the presence of a particular value in one group does not influence its presence in another group
- continuous means they are numeric and flow in the same interval of 1 unit
- all possible 4-point groups are:
(1) 0,1,2,3
(2) 1,2,3,4
(3) 2,3,4,5
(4) 3,4,5,6
(5) 4,5,6,7
(6) 5,6,7,8
(7) 6,7,8,9
(8) 7,8,9,10
Explanation:
(A) the probability that the maximum of these values is below 5 is 2/8 or 1/4
(B) the probability that the maximum of these values is between 5 and 7 is the same as the probability that the maximum of these values is 6
This probability is equal to 1/8
(C) find a formula for the probability distribution function of the 3rd highest of these values
From the 8 groups, the 3rd highest values are: 2,3,4,5,6,7,8,9
The probability distribution function of these values would enlist the probabilities of obtaining each of these values in third place, across the 8 groups
They all have the same probability which is 1/8
A formula for this PDF would be
Sigma(summation notation) from 1 to 8 of P (X1, X2, X3,...,X8)
Where X is the 3rd highest value in each group
X1 is the probability that a third highest value across the 8 groups is 2. That is, P(X=2)
Same goes for all the others
X8 is the probability that X is 9
(D) the probability that the third highest of these values is between 5 and 7 is the same as the probability that the third highest is 6
This is found in group 5 and the probability is 1/8
(E) the expectation or expected value of the third highest in each group is the mean of all third highest values.
E.V. = (2+3+4+5+6+7+8+9) รท 8
E.V. = 5.5
You are welcome.