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Chapter 5: Q9E (page 333)

A manufacturer believes that an unknown proportionP of parts produced will be defective. She models P ashaving a beta distribution.The manufacturer thinks that Pshould be around 0.05, but if the first 10 observed productswere all defective, the mean of P would rise from 0.05 to0.9. Find the beta distribution that has these properties.

Short Answer

Expert verified

The parameters of the beta distribution are \(\alpha = \frac{1}{{17}},\beta = \frac{{19}}{{17}}\).

Step by step solution

01

Defining the pdf

It is given that P is a random variable which denotes defective items in large lots. It follows beta distribution with parameters \(\alpha \,\,and\,\,\beta \).

\(f\left( p \right) = \int\limits_0^1 {{p^{\alpha - 1}}{{\left( {1 - p} \right)}^{\beta - 1}}dp} \)

02

Calculating the expectation equation

Given,

\(\begin{array}{l}E\left( p \right) = 0.05\\\end{array}\)

By the properties of a beta distribution

\(E\left( p \right) = \frac{\alpha }{{\alpha + \beta }}\)

Therefore,

\(\begin{aligned}{c}\frac{\alpha }{{\alpha + \beta }}& = 0.05\\ &= \frac{1}{{20}}\end{aligned}\)

Solving further,

\(\begin{aligned}{}20\alpha & = \alpha + \beta \\\beta& = 19\alpha \ldots \left( 1 \right)\end{aligned}\)

03

Define the theorem

By theorem 5.8.2

Suppose that P has the beta distribution with parameters \(\alpha \,\,and\,\,\beta \), and the conditional distribution ofX given P=p is the binomial distribution with parameters n and p. Then the conditional distribution of P given X=x is the beta distribution with parameters \(\alpha + x\,\,and\,\,\beta + n - x\).

04

Finding the new expectation equation

After checking first 10 observations,the expectation rises to 0.9. This is the conditional distribution mentioned in the previous step. Therefore,then the conditional distribution of P given X=x is the beta distribution with parameters \(\alpha + x\,\,and\,\,\beta + n - x\)

Here, x =10 and n=10

Therefore,

\(\begin{aligned}{}\frac{{\alpha + x}}{{\alpha + x + \beta + n - x}} &= 0.9\\\frac{{\alpha + x}}{{\alpha + \beta + n}} &= \frac{9}{{10}}\\\frac{{\alpha + 10}}{{\alpha + \beta + 10}} &= \frac{9}{{10}}\end{aligned}\)

Solving further,

\(\begin{aligned}{}10\left( {\alpha + 10} \right) &= 9\left( {\alpha + \beta + 10} \right)\\\alpha - 9\beta & = - 10 \ldots \left( 2 \right)\end{aligned}\)

05

Finding the parameter values

From (2)

\(\alpha - 9\beta = - 10\)

Substitute \(\beta = 19\alpha \) from (1)

\(\begin{aligned}{}\alpha - 9\left( {19\alpha } \right) &= - 10\\\left( {1 - 171} \right)\alpha & = - 10\\170\alpha & = 10\\\alpha & = \frac{1}{{17}}\end{aligned}\)

Using (1) and substituting value

\(\begin{array}{}\beta = 19\alpha \\ = \frac{{19}}{{17}}\end{array}\)

Therefore, \(\alpha = \frac{1}{{17}}\) and \(\beta = \frac{{19}}{{17}}\).

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Most popular questions from this chapter

Review the derivation of the Black-Scholes formula (5.6.18). For this exercise, assume that our stock price at time u in the future is

\({{\bf{S}}_{\bf{0}}}{{\bf{e}}^{{\bf{\mu u + }}{{\bf{W}}_{\bf{u}}}}}\)where \({{\bf{W}}_{\bf{u}}}\) has the gamma distribution with parameters 伪u and 尾 with 尾 > 1. Let r be the risk-free interest rate.

a. Prove that \({{\bf{e}}^{{\bf{ - ru}}}}{\bf{E}}\left( {{{\bf{S}}_{\bf{u}}}} \right){\bf{ = }}{{\bf{S}}_{\bf{0}}}\,\,{\bf{if }}\,{\bf{and}}\,\,{\bf{only}}\,\,{\bf{if}}\,{\bf{\mu = r - \alpha log}}\left( {\frac{{\bf{\beta }}}{{{\bf{\beta - 1}}}}} \right)\)

b. Assume that \({\bf{\mu = r - \alpha log}}\left( {\frac{{\bf{\beta }}}{{{\bf{\beta - 1}}}}} \right)\). Let R be 1 minus the c.d.f. of the gamma distribution with parameters 伪u and 1. Prove that the risk-neutral price for the option to buy one share of the stock for the priceq at the time u is \(\begin{array}{l}{{\bf{S}}_{\bf{0}}}{\bf{R}}\left( {{\bf{c}}\left[ {{\bf{\beta - 1}}} \right]} \right){\bf{ - q}}{{\bf{e}}^{{\bf{ - ru}}}}{\bf{R}}\left( {{\bf{c\beta }}} \right)\,\,{\bf{where}}\\{\bf{c = log}}\left( {\frac{{\bf{q}}}{{{{\bf{S}}_{\bf{0}}}}}} \right){\bf{ + \alpha ulog}}\left( {\frac{{\bf{\beta }}}{{{\bf{\beta - 1}}}}} \right){\bf{ - ru}}\end{array}\)

c. Find the price for the option being considered when u = 1, q = \({{\bf{S}}_{\bf{0}}}\), r = 0.06, 伪 = 1, and 尾 = 10.

Suppose that a fair coin is tossed until at least one head and at least one tail has been obtained. Let X denote the number of tosses that are required. Find the p.f. of X

Suppose that men arrive at a ticket counter according to a Poisson process at the rate of 120 per hour, and women arrive according to an independent Poisson process at the rate of 60 per hour. Determine the probability that four or fewer people arrive in a one-minute period.

Suppose that an electronic system contains n components that function independently of each other, and suppose that these components are connected in series, as defined in Exercise 5 of Sec. 3.7. Suppose also that each component will function properly for a certain number of periods and then will fail. Finally, suppose that for i =1,...,n, the number of periods for which component i will function properly is a discrete random variable having a geometric distribution with parameter \({p_i}\). Determine the distribution of the number of periods for which the system will function properly.

Suppose that on a certain examination in advanced mathematics, students from university A achieve scores normally distributed with a mean of 625 and a variance of 100, and students from university B achieve scores normally distributed with a mean of 600 and a variance of 150. If two students from university A and three students from university B take this examination, what is the probability that the average of the scores of the two students from university A will be greater than the average of the scores of the three students from university B? Hint: Determine the distribution of the difference between the two averages
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