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Chapter 14: Problem 19

Let \(X\) be a Gamma r.v. with parameters \((\alpha, \beta)\). One can calculate its characteristic function without using contour integration. Assume \(\beta=1\) and expand \(e^{i x}\) in a power series. Then show $$ \frac{1}{\Gamma(\alpha)} \sum_{n=1}^{\infty} \frac{(i u)^{n}}{n !} \int_{0}^{\infty} e^{-x} x^{n+\alpha-1} d x=\sum_{n=0}^{\infty} \frac{\Gamma(n+\alpha)}{n ! \Gamma(\alpha)}(i u)^{n} $$ and show this is a binomial series which sums to \(\frac{1}{(1-i u)^{\alpha}}\).

Short Answer

Expert verified
The characteristic function simplifies to \(\varphi_X(u) = \frac{1}{(1-iu)^\alpha}\).

Step by step solution

01

Understand the Setup

The characteristic function of a Gamma random variable is given by \[ \varphi_X(u) = E[e^{iuX}] = \int_0^\infty e^{iux} \frac{x^{\alpha-1}e^{-x/\beta}}{\beta^\alpha \Gamma(\alpha)} \, dx \]Here, we are asked to evaluate it without contour integration and consider \(\beta = 1\). This simplifies the integrand to \(e^{ix} x^{\alpha-1}e^{-x}\). Expand \(e^{ix}\) in a power series.
02

Expansion of Exponential in Power Series

Expand the exponential term \(e^{iux}\) using Taylor series:\[ e^{iux} = \sum_{n=0}^{\infty} \frac{(iux)^n}{n!} \]Plug this series into the integral:\[ \int_0^\infty e^{ix} x^{\alpha-1}e^{-x} \, dx = \int_0^\infty \sum_{n=0}^{\infty} \frac{(iux)^n}{n!} x^{\alpha-1}e^{-x} \, dx \]Now interchange the sum and integral due to Fubini's theorem.
03

Interchange Sum and Integral

Interchanging sum and integral gives:\[ \sum_{n=0}^{\infty} \frac{(iu)^n}{n!} \int_0^\infty x^{n+\alpha-1} e^{-x} \, dx \]This integral is the definition of the Gamma function:\[ \Gamma(n+\alpha) = \int_0^\infty x^{n+\alpha-1} e^{-x} \, dx \]
04

Rewrite the Summation

Substituting the expression for the Gamma function:\[ \sum_{n=0}^{\infty} \frac{(iu)^n \Gamma(n+\alpha)}{n!} \]Use the original factor \(\frac{1}{\Gamma(\alpha)}\) which is from the PDF of the Gamma distribution:\[ \varphi_X(u) = \sum_{n=0}^{\infty}\frac{\Gamma(n+\alpha)}{n! \Gamma(\alpha)}(iu)^n \]
05

Recognizing the Binomial Series

Recognize that with exponent \(\alpha\), the series \(\sum_{n=0}^{\infty} \frac{\Gamma(n+\alpha)}{n! \Gamma(\alpha)} (iu)^n\) is a binomial series of the form: \[ \frac{1}{(1-iu)^\alpha} = \sum_{n=0}^{\infty} \binom{-\alpha}{n}(-iu)^n \] This identifies as the binomial series expansion.
06

Conclusion

Hence, we conclude that the characteristic function sums as:\[ \varphi_X(u) = \frac{1}{(1-iu)^\alpha} \]Thus, the characteristic function of a Gamma random variable simplifies to this rational form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Characteristic Function
A characteristic function, often denoted as \( \varphi_X(u) \), is a mathematical tool used to study random variables. It is defined as the expectation \( E[e^{iuX}] \) of the complex exponential \( e^{iuX} \), where \( i \) is the imaginary unit, and \( u \) is a real number.
For a random variable \( X \), this function helps encapsulate all the distributional information in a way that is easy to manipulate in proofs and calculations.
Characteristic functions are unique to the probability distribution of the random variable, meaning different distributions have different characteristic functions.
  • They are often utilized to derive moments of the distribution, as derivatives of the function at zero provide these moments.
  • Characteristic functions are crucial in theoretical probability, especially in the study of convergence and the central limit theorem.
  • They offer a simplified way to handle integration and product operations through the exponential's properties.
Power Series Expansion
The power series expansion is a method of expressing functions like \( e^{iux} \) as infinite sums of terms. This happens through the Taylor series, where each term of the series is a derivative of the function calculated at zero, divided by the factorial of the term number.
For the function \( e^{iux} \), it expands as:
\[ e^{iux} = \sum_{n=0}^{\infty} \frac{(iux)^n}{n!} \]
Power series expansions are quite useful in calculus and analysis, because they allow functions to be expressed as sums of polynomials, which are often easier to handle.
  • They help to analyze functions over a range of values, especially near the center of convergence.
  • This method is particularly powerful for complex numbers as it brings simplicity to complicated transcendental functions.
  • Using power series allows for easy differentiation and integration term by term, greatly simplifying many mathematical problems.
Binomial Series
The binomial series is an infinite series expansion of the binomial expression \((1-x)^k\), where \( k \) is not necessarily an integer. It provides a way to expand expressions raised to any power in terms of a series:
\[ (1-x)^k = \sum_{n=0}^{\infty} \binom{k}{n} (-x)^n \]
In the context of the characteristic function of a Gamma variable, this expansion becomes essential. It helps identify that the function can be rewritten as a binomial series, showing that the function can be simplified:
\[ (1-iu)^{-\alpha} = \sum_{n=0}^{\infty} \frac{\Gamma(n+\alpha)}{n! \Gamma(\alpha)} (iu)^n \]
  • This series is useful for approximating expressions and converting them to a form that can be easily used for calculations and proofs.
  • It shows the versatility of binomial coefficients in solving real-world problems.
  • The convergence of the binomial series depends on the value of \( x \) and \( k \), indicating its importance in convergence analysis.
Gamma Function
The Gamma function, denoted \( \Gamma(\alpha) \), extends the factorial function to complex and real number arguments. For positive integers, it simplifies to the factorial, namely \( \Gamma(n) = (n-1)! \). However, its use for non-integer values is invaluable in continuous mathematics.
The Gamma function definition is given by:
\[ \Gamma(\alpha) = \int_0^{\infty} x^{\alpha-1} e^{-x} \, dx \]
This function plays a central role in the probability distributions such as the Gamma distribution.
  • It is used to solve integrals that appear in distribution equations, especially those involving exponentials and powers.
  • Gamma functions often appear in the normalization factors of probability distributions.
  • The function's properties, like \( \Gamma(\alpha+1) = \alpha \Gamma(\alpha) \), are powerful tools for calculating complex integrals and expressions.
Understanding the Gamma function helps in mastering many concepts in advanced mathematics and statistical distributions.

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Most popular questions from this chapter

Let \(X\) be a positive random variable. The Mellin transform of \(X\) is defined to be $$ T_{X}(\theta)=E\left\\{X^{\theta}\right\\} $$ for all values of \(\theta\) for which the expected value of \(X^{\theta}\) exists. a) Show that $$ T_{X}(\theta)=\varphi_{\log X}\left(\frac{\theta}{i}\right) $$ when all terms are well defined. b) Show that if \(X\) and \(Y\) are independent and nonnegative, then $$ T_{X Y}(\theta)=T_{X}(\theta) T_{Y}(\theta) $$ c) Show that \(T_{b X^{a}}(\theta)=b^{\theta} T_{X}(a \theta)\) for \(b>0\) and \(a \theta\) in the domain of definition of \(T_{X}\).

Let \(f(x)=\frac{1}{\pi\left(1+x^{2}\right)}\), a Cauchy density, for a r.v. \(X\). Show that $$ \varphi_{X}(u)=e^{-|u|} $$ by integrating around a semicircle with diameter \([-R, R]\) on the real axis to the left, from the point \((-R, 0)\) to the point \((R, 0)\) over the real axis.

Let \((X, Y)\) be independent normals, both with means \(\mu=0\) and variances \(\sigma^{2}\). Let $$ Z=\sqrt{X^{2}+Y^{2}} \text { and } W=\operatorname{Arctan}\left(\frac{X}{Y}\right), \quad-\frac{\pi}{2}

Let \(X\) be \(N(0,1)\). Show that \(E\left\\{X^{2 n+1}\right\\}=0\) and $$ E\left\\{X^{2 n}\right\\}=\frac{(2 n) !}{2^{n} n !}=(2 n-1)(2 n-3) \ldots 3 \cdot 1 . $$

Let \(X\) be \(N(0,1)\). Let $$ M(s)=E\left\\{e^{s X}\right\\}=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}} \exp \left(s x-\frac{1}{2} x^{2}\right) d x $$ Show that \(M(s)=e^{s^{2} / 2}\).
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