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Chapter 8: Problem 8

Beta-Darstellung der Binomialverteilung. Zeigen Sie durch Differentiation nach \(p:\) F眉r alle \(0

Short Answer

Expert verified
The binomial probability sum equals the Beta function integral over [0, p].

Step by step solution

01

Understand the Problem

We need to show that the sum of probabilities from \(k+1\) to \(n\), denoted as \(\mathscr{B}_{n,p}(\{k+1, k+2, \ldots, n\})\), can be expressed as a certain integral involving Beta functions. This includes demonstrating the equivalency between the given binomial probability configuration and a Beta probability integral over the interval \([0, p]\).
02

Recall Binomial Probability Sum

For a binomial distribution, the probability of getting at least \(k+1\) successes out of \(n\) trials is given by the cumulative binomial distribution function from \(k+1\) to \(n\). This is expressed as a sum of terms, \[ \mathscr{B}_{n,p}(\{k+1, k+2, \ldots, n\}) = \sum_{j=k+1}^{n} \binom{n}{j} p^j (1-p)^{n-j} \].
03

Set Up the Integral Representation

Recognize that the integral \( \int_{0}^{p} t^{k}(1-t)^{n-k-1} dt \) represents the cumulative distribution function of the Beta distribution. The Beta distribution function \( \beta_{k+1, n-k}([0,p]) \) is evaluated over the interval \([0, p]\), which matches the format of our problem.
04

Differentiation by Parts

To solve the given problem using differentiation, we apply the integral of Beta function, \[ B(x; a, b) = \int_{0}^{x} t^{a-1}(1-t)^{b-1} dt \]. We differentiate this with respect to \(p\) to express it in terms of \(n\), \(k\), and \(p\). After applying the Fundamental Theorem of Calculus, we find:\[ \frac{d}{dp} B(p; k+1, n-k) = p^{k}(1-p)^{n-k-1} \].
05

Understand the Resulting Expression

Integrate the differentiated expression to account for the interval \([0, p]\), thus relating it to \(\beta_{k+1, n-k}([0, p])\), and express it as the coefficient involved, which matches the binomial coefficient representation. This reflects a connection formed by completing the integral evaluation for Beta function and binomial identities.
06

Synthesis of Formulaic Equivalence

Combine the integral representation with the binomial coefficients:\[ \left( \begin{array}{c} n \ k \end{array} \right)(n-k) = \beta_{k+1, n-k}([0, p]) \].Thus, the mathematical expression translates the summation of binomial terms over \(p\) into a Beta distribution function over the same interval.

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Key Concepts

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

Binomial Distribution
The binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent trials. Each trial results in either success or failure. The probability of success in each trial is denoted by \( p \), while the number of trials is \( n \). The probability of getting exactly \( k \) successes in \( n \) trials is given by the formula:
  • \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)
  • Here, \( \binom{n}{k} \) is the binomial coefficient, which counts the number of ways to choose \( k \) successes out of \( n \) trials.
  • The term \( p^k (1-p)^{n-k} \) represents the probability of exactly \( k \) successes and \( n-k \) failures.
By utilizing the binomial distribution, we can calculate various probabilities, such as the likelihood that an event occurs a certain number of times. This is foundational in probability theory and is widely used in statistical applications.
Beta Function
The Beta function is a special function in mathematics that appears in numerous probability and calculus applications. It is defined as an integral for positive \( x \) and \( y \):
  • \( B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt \)
  • This integral involves the product of power functions \( t^{x-1} \) and \( (1-t)^{y-1} \).
The connection between the Beta function and distributions comes from its role in defining beta distribution, which is important for modeling random variables limited to intervals like \([0,1]\). Furthermore, it is related to binomial distributions via combinatorial identities, providing a continuous analogue for discrete distributions.
Integral Representation
An integral representation refers to expressing a mathematical object, such as a function or distribution, using an integral. In the context of probability theory, this is often used to transition from discrete to continuous formulations.
  • The exercise involves representing a sum of binomial probabilities as an integral over a continuous range \([0, p]\).
  • The integral \( \int_0^p t^k (1-t)^{n-k-1} dt \) provides the continuous form.
This transformation simplifies certain computations or comparisons, especially when dealing with distributions. Integral expressions can bridge between point probabilities in discrete systems and area under curves in continuous distributions.
Differentiation
Differentiation is a fundamental mathematical tool used to find the rate at which a function changes. It is crucial in both calculus and probability theory for analyzing changes and solving equations.
  • In the given exercise, differentiation is used to transition from the Beta function integral to variable terms, particularly \( p \).
  • This is achieved by differentiating the integral \( B(p; k+1, n-k) \) with respect to \( p \).
By applying differentiation, we can derive the probability density functions from cumulative distribution functions. This linkage is essential for detailed probability analyses.
Cumulative Distribution Function
A cumulative distribution function (CDF) provides the probability that a random variable is less than or equal to a certain value. It adds up probabilities from a probability distribution.
  • The CDF of a binomial distribution provides the probability of obtaining \( k \) or fewer successes in \( n \) trials.
  • Mathematically, a CDF is represented as: \[ F(x) = P(X \leq x) \]
  • For the Beta function, its cumulative form is expressed through an integral that evaluates this probability over a range \([0, p]\).
The CDF is a vital tool because it summarizes the distribution of a random variable. It helps in understanding all the probabilities up to a certain point, serving as a holistic measure of the variable's distribution.

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Most popular questions from this chapter

Zur Geschichte der EU: Die FAZ vom \(23.06 .1992\) berichtete, dass \(26 \%\) der Deutschen mit einer einheitlichen europaischen W?hrung einverstanden w盲ren; ferner seien \(50 \%\) f眉r eine 脰ffnung der EU nach Osten. Die Zahlenwerte basierten auf einer Allensbach-Umfrage unter rund 2200 Personen. W盲ren genauere Prozentangaben (also z. B. mit einer Stelle nach dem Komma) sinnvoll gewesen? Betrachten Sie hierzu die L盲nge der approximativen Konfidenzintervalle zum Irrtumsniveau \(0.05 !\)

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Mehrdeutigkeit von Quantilen. Sei \(Q\) ein Wahrscheinlichkeitsma脽 auf \((\mathbb{R}, \mathscr{B}), 0<\) \(\alpha<1, q\) ein \(\alpha\)-Quantil von \(Q\), und \(q^{\prime}>q .\) Zeigen Sie: Genau dann ist auch \(q^{\prime}\) ein \(\alpha\)-Quantil von \(Q\), wenn \(Q(] q, q^{\prime}[)=0\)

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Sensitivit?t von Stichprobenmittel, -median und getrimmten Mitteln. Sei \(T_{n}: \mathbb{R}^{n} \rightarrow \mathbb{R}\) eine Statistik, welche \(n\) reellen Beobachtungswerten einen,,Mittelwert" zuordnet. Wie stark \(T_{n}\) bei gegebenen Werten \(x_{1}, \ldots, x_{n-1} \in \mathbb{R}\) von einer Einzelbeobachtung \(x \in \mathbb{R}\) abh盲ngt, wird beschrieben durch die Sensitivit盲tsfunktion $$ S_{n}(x)=n\left(T_{n}\left(x_{1}, \ldots, x_{n-1}, x\right)-T_{n-1}\left(x_{1}, \ldots, x_{n-1}\right)\right) $$ Bestimmen und zeichnen Sie \(S_{n}\) in den Fallen, wenn \(T_{n}\) (a) der Stichprobenmittelwert, (b) der Stichprobenmedian, und (c) das \(\alpha\)-getrimmte Mittel $$ \left(x_{\lfloor n \alpha\rfloor+1: n}+\cdots+x_{n-\lfloor n \alpha\rfloor: n}\right) /(n-2\lfloor n \alpha\rfloor) $$ zu einem Trimm-Niveau \(0 \leq \alpha<1 / 2\) ist. Hier bezeichnet \(x_{k: n}\) die \(k\)-te Ordnungsstatistik von \(x_{1}, \ldots, x_{n}\)
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