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Chapter 7: Q4E (page 425)

Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from the Bernoulli distribution with parameter 胃, which is unknown, but it is known that 胃 lies in the open interval 0 <胃< 1. Show that the M.L.E. of 胃 does not exist if every observed value is 0 or if every observed value is 1.

Short Answer

Expert verified

M.L.E. of 胃 does not exist if every observed value is 0 or if every observed value is 1.

Step by step solution

01

Given information

\({X_1},...,{X_n}\) form a random sample from the Bernoulli distribution with parameter 胃,We need to prove that the M.L.E. of 胃 does not exist if every observed value is 0 or if every observed value is 1.

02

Proof of the M.L.E. of θ does not exist if every observed value is 0 or if every observed value is 1.

Let S be the sum of observations in the sample. The parameter \(\theta \) follows Bernoulli distribution. If S=0, then S is a decreasing function of \(\theta \).

Again if S=n then S is a increasing function of \(\theta \) .Neither 0 nor 1 is in the parameter space of \(\theta \) which indicates no such M.L.E exists.

Hence the proof.

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Most popular questions from this chapter

Suppose that the proportion 胃 of defective items in a large shipment is unknown and that the prior distribution of 胃 is the beta distribution with parameters 2 and 200. If 100 items are selected at random from the shipment and if three of these items are found to be defective, what is the posterior distribution of 胃?

Consider the data in Example 7.3.10. This time, suppose that we use the improper prior 鈥減.d.f.鈥漒(\xi \left( \theta \right) = 1\)(for all 胃). Find the posterior distribution of\(\theta \)and the posterior probability that\(\theta > 1\).

Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which the mean \(\lambda \)is either 1.0 or 1.5, and the prior p.f. of \(\lambda \) is as follows:

\(\xi \left( {1.0} \right) = 0.4\)and\(\xi \left( {1.5} \right) = 0.6\)

If a roll of tape selected at random is found to have three defects, what is the posterior p.f. of \(\lambda \) ?

Question:It is not known what proportion p of the purchases of a certain brand of breakfast cereal are made by women and what proportion are made by men. In a random sample of 70 purchases of this cereal, it was found that 58 were made by women and 12 were made by men. Find the M.L.E. of p.

Suppose that the number of defects in a 1200-foot roll of magnetic recording tape has a Poisson distribution for which the value of the mean 胃 is unknown and that the prior distribution of 胃 is the gamma distribution with parameters 伪 = 3 and 尾 = 1. When five rolls of this tape are selected at random and inspected, the numbers of defects found on the rolls are 2, 2, 6, 0, and 3. Determine the posterior distribution of 胃.
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