Description
a. If any of the observations is greater than N, then there is no doubt that the
process is Type II.
b. If none of the observations is greater than N, then the process can be either Type I
or Type II. The probability of Type I is always more than 50%. If M is close to N,
then the probability of Type II will approach from below 50%. If M tends to
infinite, then the probability of Type II tends to zero.
2.
a.
i. P(X1 ≤ N, …, XK ≤ N | Type I) = (P(Xi ≤ N) | Type I)K = 1. As seen, if the
sample is obtained from Type 1, it is a sure event that all of the
observations are less than or equal to N.
ii. P(X1 ≤ N, …, XK ≤ N | Type II) = (P(Xi ≤ N) | Type II)K = (N/M)K
. In the second
case, that probability corresponds to a binomial distribution with
parameters p=N/M and n=K. In this case this is equivalent to have K
successes in K trials.
b. Bayes Theorem:
pg. 2
i. P(Type I | X1 ≤ N, …, XK ≤ N) =
?(???? ?)∗?(?1 ≤ N,…,?? ≤ N |???? ?)
?(???? ?)∗?(?1 ≤ N,…,?? ≤ N |???? ?)+?(???? ??)∗?(?1 ≤ N,…,?? ≤ N |???? ??)
=
(1/2)(1)
(
1
2
)(1)+(
1
2
)(
?
?
)
?
=
1
1+(
?
?
)
?
ii. P(Type I | X1 ≤ N, …, XK ≤ N) = 1 –
1
1+(
?
?
)
?
We can appreciate how this conditional probabilities correspond to the
intuitions in question 1. If M tends to N, then P(Type II) tends to ½. If M
tends to infinite, P(Type II) tends to 0.
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