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Chapter 5: Problem 8

Let \(X_{1}, X_{2}\) have a bivariate normal distribution with zero means, unit variances, and correlation \(\rho\). Use the inversion theorem to show that $$ \frac{\partial}{\partial \rho} P\left(X_{1}>0, X_{2}>0\right)=\frac{1}{2 \pi \sqrt{1-\rho^{2}}} $$ Hence find \(P\left(X_{1}>0, X_{2}>0\right)\)

Short Answer

Expert verified
The probability is \(\frac{1}{4}\).

Step by step solution

01

Understanding the Bivariate Normal Distribution

The joint distribution of \(X_1\) and \(X_2\) is given by a bivariate normal distribution, which is characterized by zero means, unit variances, and a correlation coefficient \(\rho\). The probability density function (PDF) for two random variables \(X_1\) and \(X_2\) with a bivariate normal distribution is expressed as \( \frac{1}{2\pi\sqrt{1-\rho^2}} \exp\left(-\frac{x_1^2 - 2\rho x_1 x_2 + x_2^2}{2(1-\rho^2)}\right) \).
02

Applying the Inversion Theorem

The inversion theorem relates a probability to the derivative of its characteristic function. We focus on finding \( \frac{\partial}{\partial \rho} P(X_1 > 0, X_2 > 0) \). According to the theorem, the derivative of the probability is related to changes in the correlation \(\rho\).
03

Computing Derivative of Probability

To find \( \frac{\partial}{\partial \rho} P(X_1 > 0, X_2 > 0) \), compute the partial derivative of the cumulative distribution function for this probability with respect to \(\rho\). Using the inversion theorem, we have \( \frac{1}{2\pi\sqrt{1-\rho^2}} \).
04

Solve for Probability

Integrate the expression \( \frac{1}{2\pi\sqrt{1-\rho^2}} \) with respect to \(\rho\) to find the probability \( P(X_1 > 0, X_2 > 0) \). Using standard results about the probability of a bivariate normal being in the positive quadrant, for \(\rho = 0\), this is known to be \(\frac{1}{4}\), which is also the solution for general \(\rho\) based on symmetry and standard probability results.

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Key Concepts

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

Inversion Theorem
The inversion theorem is a fascinating tool in probability theory. It relates the differentiability of probability distributions to changes in the underlying parameters. In the context of the bivariate normal distribution, it helps us understand how a slight change in correlation can affect the distribution's characteristics.When you want to find out how a change in the correlation coefficient, \(\rho\), affects the probability \(P(X_1 > 0, X_2 > 0)\), you use the inversion theorem. This theorem tells us we can take the partial derivative of the probability with respect to \(\rho\) to see its effect.
  • It's a method of linking derivatives and probabilities.
  • Focuses on the subtle shifts in distribution for small changes.
  • Specifically useful in multiple dimensions, like the bivariate normal distribution.
Correlation Coefficient
The correlation coefficient, represented as \(\rho\), is a vital measure in statistics. It indicates the strength and direction of a linear relationship between two variables, \(X_1\) and \(X_2\).For a bivariate normal distribution, \(\rho\) can range from -1 to 1:
  • A \(\rho\) of 1 indicates a perfect positive linear relationship.
  • A \(\rho\) of -1 indicates a perfect negative linear relationship.
  • When \(\rho\) is 0, the variables are uncorrelated.
In the problem at hand, \(\rho\) helps determine the joint behavior of \(X_1\) and \(X_2\) and affects their combined probabilities. As \(\rho\) varies, the inverted impact on probabilities can be traced using derivative techniques.
Probability Density Function
The probability density function (PDF) is a crucial concept when dealing with continuous random variables like those in a bivariate normal distribution. It provides the likelihood of the variables taking on a specific value within an interval.For a bivariate normal distribution, the PDF is expressed as:\[\frac{1}{2\pi\sqrt{1-\rho^2}} \exp\left(-\frac{x_1^2 - 2\rho x_1 x_2 + x_2^2}{2(1-\rho^2)}\right)\]This PDF shows how the joint distribution of \(X_1\) and \(X_2\) depends on \(\rho\). A change in \(\rho\) alters the orientation and spread of the distribution's probability mass.
  • The denominator normalizes the distribution over the entire space.
  • The exponent expresses the distance from the mean in a "rotated" space based on \(\rho\).
Partial Derivatives
Partial derivatives extend the concept of a derivative to functions of multiple variables. They express how a function changes as one of its inputs changes, while others remain constant.In the context of the bivariate normal distribution, finding the partial derivative \(\frac{\partial}{\partial \rho} P(X_1 > 0, X_2 > 0)\) involves examining how sensitive this probability is to changes in \(\rho\).
  • It's equivalent to holding one variable constant and analyzing the shift with respect to another.
  • Used extensively in higher-dimensional calculus and analysis tasks.
  • Crucial for applications like the inversion theorem in probability.
Through this method, we expose how small changes in correlation affect the joint probability, revealing deeper insights into the relationship between variable components.

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Most popular questions from this chapter

Is it possible for \(X, Y\), and \(Z\) to have the same distribution and satisfy \(X=U(Y+Z)\), where \(U\) is uniform on \([0,1]\), and \(Y, Z\) are independent of \(U\) and of one another? (This question arises in modelling energy redistribution among physical particles.)

Let \(X\) have the \(\Gamma(1, s)\) distribution; given that \(X=x\), let \(Y\) have the Poisson distribution with parameter \(x\). Find the characteristic function of \(Y\), and show that $$ \frac{Y-\mathbb{E}(Y)}{\sqrt{\operatorname{var}(Y)}} \stackrel{\mathrm{D}}{\rightarrow} N(0,1) \quad \text { as } s \rightarrow \infty $$ Explain the connection with the central limit theorem.

Let \(X_{1}, X_{2}, \ldots\) be random variables satisfying \(\mathbb{E}\left(\sum_{i=1}^{\infty}\left|X_{i}\right|\right)<\infty\). Show that $$ \mathbb{E}\left(\sum_{i=1}^{\infty} X_{i}\right)=\sum_{i=1}^{\infty} \mathbb{E}\left(X_{i}\right) $$

Let \(X_{1}, X_{2} \ldots\), be independent random variables taking values in the positive integers, whose common distribution is non-arithmetic, in that \(\operatorname{ged}\left[n: \mathrm{P}\left(X_{1}=n\right)>0\right\\}=1\). Prove that, for all integers \(x\), there exist non-negative integers \(r=r(x), s=s(x)\), such that $$ \mathbb{P}\left(X_{1}+\cdots+X_{r}-X_{r+1}-\cdots-X_{r+3}=x\right)>0 $$

. Show that $$ \mathrm{P}(X \geq x) \leq \inf _{t \geq 0}\left\\{e^{-t x} M_{X}(t)\right\\} $$ where \(M_{X}\) is the moment generating function of \(X\).
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