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Chapter 7: Q3E (page 425)

Question: Consider again the conditions in Exercise 2, but suppose also that it is known that\(\)\(\frac{{\bf{1}}}{{\bf{2}}} \le {\bf{p}} \le \frac{{\bf{2}}}{{\bf{3}}}\). If the observations in the random sample of 70 purchases are as given in Exercise 2, what is the M.L.E. of p?

Short Answer

Expert verified

\(p = \frac{2}{3}\)

Step by step solution

01

Given information

In the exercise 2 the sample size is 70,\(\frac{1}{2} \le p \le \frac{2}{3}\) we need to find out the M.L.E. of p.

02

Step-2: Calculation of M.L.E. of p

We are given that \(\frac{1}{2} \le p \le \frac{2}{3}\)and we are interested to determine the value of p as maximum likelihood estimator. The maximum likelihood estimator function of p follows Bernoulli distribution with the probability mass function\(f\left( x \right) = {p^x}{\left( {1 - p} \right)^{n - x}}\) where p denote probability of success .Here x= 58 and n= 70. The function along with the given interval is maximum when \(p = \frac{2}{3}\).

So, the required M.L.E. is \(p = \frac{2}{3}\)

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Most popular questions from this chapter

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from the gamma distribution specified in Exercise 6. Show that the statistic \({\bf{T = }}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{\bf{log}}{{\bf{x}}_{\bf{i}}}} \) is a sufficient statistic for the parameter\({\bf{\alpha }}\).

Suppose that a random sample \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) is drawn from the Pareto distribution with parameters \({{\bf{x}}_{\bf{0}}}\,\,\,\,{\bf{and}}\,\,\,{\bf{\alpha }}\).

a. If \({{\bf{x}}_{\bf{0}}}\) is known and \({\bf{\alpha > 0}}\) unknown, find a sufficient statistic

b. If \({\bf{\alpha }}\) is known and \({{\bf{x}}_{\bf{0}}}\) unknown, find a sufficient statistic.

Assume that the random variables \({X_1},...{X_n}\) form a random sample of size n from the distribution specified in that exercise, and show that the statistic T specified in the exercise is a sufficient statistic for the parameter.

1. The negative binomial distribution with parameters r and p where r is known and p is unknown. \(\left( {0 < p < 1} \right)\)\(T = \sum\limits_{i = 1}^n {{X_i}} \).

Consider a distribution for which the pdf. or the p.f. is \(f\left( {x|\theta } \right)\) , where the parameter 胃 is a k-dimensional vector belonging to some parameter space\(\Omega \) . It is said that the family of distributions indexed by the values of 胃 in\(\Omega \) is a k-parameter exponential family, or a k-parameter Koopman-Darmois family, if \(f\left( {x|\theta } \right)\)can be written as follows for \(\theta \in \Omega \)and all values of x:

\(f\left( {x|\theta } \right) = a\left( \theta \right)b\left( x \right)\exp \left( {\sum\limits_{i = 1}^k {{c_i}\left( \theta \right){d_i}\left( x \right)} } \right)\)

Here, a and \({c_1},...,{c_k}\) are arbitrary functions of 胃, and b and \({d_1},...,{d_k}\) are arbitrary functions of x. Suppose now that \({X_1},...,{X_n}\) form a random sample from a distribution which belongs to a k-parameter exponential family of this type, and define the k statistics \({T_1},...,{T_k}\) as follows:

\({T_i} = \sum\limits_{j = 1}^n {{d_i}\left( {{X_j}} \right)} \)

Show that the statistics \({T_1},...,{T_k}\)are jointly sufficient statistics for 胃.

Let 尉(胃)be a p.d.f. that is defined as follows for constants

伪 >0 and尾 >0:

\(\xi \left( \theta \right) = \left\{ \begin{aligned}{l}\frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}{\theta ^{ - \left( {\alpha + 1} \right)}}{e^{ - \beta /\theta }}\,for\,\theta > 0\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\theta \le 0\end{aligned} \right.\)

A distribution with this p.d.f. is called an inverse gamma distribution.

a. Verify that尉(胃)is actually a p.d.f. by verifying that

\(\int_0^\infty {\xi \left( \theta \right)} d\theta = 1\)

b. Consider the family of probability distributions that can be represented by a p.d.f.尉(胃)having the given form for all possible pairs of constants伪 >0 and尾 >

0. Show that this family is a conjugate family of prior distributions for samples from a normal distribution with a known value of the mean渭and an unknown

value of the variance胃.
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