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Chapter 5: Q5E (page 333)

Suppose that\(X\)has the beta distribution with parameters\(\alpha \)and\(\beta \), and let\(r\)and\(s\)be given positive integers. Determine the value of\(E\left[ {{X^r}{{\left( {1 - X} \right)}^s}} \right]\)

Short Answer

Expert verified

The value of \(E\left[ {{X^r}{{\left( {1 - X} \right)}^s}} \right]\)is \(\frac{{\left[ {\alpha \left( {\alpha + 1} \right)...\left( {\alpha + r - 1} \right)} \right]\left[ {\beta \left( {\beta + 1} \right)...\left( {\beta + s - 1} \right)} \right]}}{{\left( {\alpha + \beta } \right)\left( {\alpha + \beta + 1} \right)...\left( {\alpha + \beta + r + s - 1} \right)}}\)

Step by step solution

01

Given information

X be the random variable follows a beta distribution with parameter \(\alpha \) and \(\beta \)

02

Given information

Pdf of beta distribution with parameters\(\alpha ,\beta > 0\)is:

\(f\left( {x|\alpha ,\beta } \right) = \left\{ \begin{array}{l}\frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}{x^{\alpha - 1}}{\left( {1 - x} \right)^{\beta - 1}}\;\;\;\;\;for\,0 < x < 1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Otherwise\end{array} \right.\)

Therefore

\(E\left[ {{X^r}{{\left( {1 - X} \right)}^s}} \right] = \int\limits_0^1 {{X^r}{{\left( {1 - X} \right)}^s}\frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}{x^{\alpha - 1}}{{\left( {1 - x} \right)}^{\beta - 1}}dx} \)\(\begin{aligned}{c}E\left[ {{X^r}{{\left( {1 - X} \right)}^s}} \right] &= \frac{{\left|\!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}\int\limits_0^1 {{x^{\alpha + r - 1}}{{\left( {1 - x} \right)}^{\beta + s - 1}}dx} \\& = \frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }} \times \frac{{\left| \!{\overline {\, {\left( {\alpha + r} \right)} \,}} \right. \left| \!{\overline {\, {\left( {\beta + s} \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( {\alpha + \beta + r + s} \right)} \,}} \right. }}\end{aligned}\)

Therefore

\(E\left[ {{X^r}{{\left( {1 - X} \right)}^s}} \right] = \frac{{\left| \!{\overline {\, {\left( {\alpha + r} \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. }} \times \frac{{\left| \!{\overline {\, {\left( {\beta + s} \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }} \times \frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( {\alpha + \beta + r + s} \right)} \,}} \right. }}\)

\(E\left[ {{X^r}{{\left( {1 - X} \right)}^s}} \right] = \frac{{\left[ {\alpha \left( {\alpha + 1} \right)...\left( {\alpha + r - 1} \right)} \right]\left[ {\beta \left( {\beta + 1} \right)...\left( {\beta + s - 1} \right)} \right]}}{{\left( {\alpha + \beta } \right)\left( {\alpha + \beta + 1} \right)...\left( {\alpha + \beta + r + s - 1} \right)}}\)

The value of \(E\left[ {{X^r}{{\left( {1 - X} \right)}^s}} \right]\)is \(\frac{{\left[ {\alpha \left( {\alpha + 1} \right)...\left( {\alpha + r - 1} \right)} \right]\left[ {\beta \left( {\beta + 1} \right)...\left( {\beta + s - 1} \right)} \right]}}{{\left( {\alpha + \beta } \right)\left( {\alpha + \beta + 1} \right)...\left( {\alpha + \beta + r + s - 1} \right)}}\)

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Most popular questions from this chapter

Let \({{\bf{X}}_{{\bf{1,}}}}{\bf{ }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)be i.i.d. random variables having the normal distribution with mean \({\bf{\mu }}\) and variance\({{\bf{\sigma }}^{\bf{2}}}\). Define\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \) , the sample mean. In this problem, we shall find the conditional distribution of each \({{\bf{X}}_{\bf{i}}}\)given\(\overline {{{\bf{X}}_{\bf{n}}}} \).

a.Show that \({{\bf{X}}_{\bf{i}}}\)and\(\overline {{{\bf{X}}_{\bf{n}}}} \) have the bivariate normal distribution with both means \({\bf{\mu }}\), variances\({{\bf{\sigma }}^{\bf{2}}}{\rm{ }}{\bf{and}}\,\,\frac{{{{\bf{\sigma }}^{\bf{2}}}}}{{\bf{n}}}\),and correlation\(\frac{{\bf{1}}}{{\sqrt {\bf{n}} }}\).

Hint: Let\({\bf{Y = }}\sum\limits_{{\bf{j}} \ne {\bf{i}}} {{{\bf{X}}_{\bf{j}}}} \).

Now show that Y and \({{\bf{X}}_{\bf{i}}}\) are independent normal and \({{\bf{X}}_{\bf{n}}}\)and \({{\bf{X}}_{\bf{i}}}\) are linear combinations of Y and \({{\bf{X}}_{\bf{i}}}\) .

b.Show that the conditional distribution of \({{\bf{X}}_{\bf{i}}}\) given\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\overline {{{\bf{x}}_{\bf{n}}}} \)\(\) is normal with mean \(\overline {{{\bf{x}}_{\bf{n}}}} \) and variance \({{\bf{\sigma }}^{\bf{2}}}\left( {{\bf{1 - }}\frac{{\bf{1}}}{{\bf{n}}}} \right)\).

Consider again the electronic system described in Exercise 10, but suppose now that the system will continue to operate until two components have failed. Determine the mean and the variance of the length of time until the system fails.

Suppose that X and Y are independent random variables, X has the gamma distribution with parameters α1 and β, and Y has the gamma distribution with parameters α2 and β. Let U = X/ (X + Y) and V = X + Y. Show that (a) U has the beta distribution with parameters α1 and α2, and (b) U and V are independent.

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 αu and β with β > 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 αu 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, α = 1, and β = 10.

Prove Corollary 5.9.2.
See all solutions

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