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Chapter 17: Problem 695

Show that if \((\mathrm{X}, \mathrm{Y})\) has a bivariate normal distribution, then the marginal distributions of \(\mathrm{X}\) and \(\mathrm{Y}\) are univariate normal distributions; that is, \(\mathrm{X}\) is normally distributed with mean \(\mu_{\mathrm{x}}\) and variance \(\sigma^{2} \mathrm{x}\) and \(\mathrm{Y}\) is normally distributed with mean \(\mu_{\mathrm{y}}\) and variance \(\sigma^{2} \mathrm{y}\).

Short Answer

Expert verified
To show that the marginal distributions of X and Y are univariate normal distributions, we first obtain the joint probability density function of the bivariate normal distribution. Then we find the marginal PDFs by integrating the joint PDF with respect to y for X and with respect to x for Y. After simplifying the integrations, we find that \(f_X(x) = \frac{1}{\sqrt{2\pi}\sigma_x} \exp\left(-\frac{1}{2}\frac{(x-\mu_x)^2}{\sigma^2_x}\right)\) and \(f_Y(y) = \frac{1}{\sqrt{2\pi}\sigma_y} \exp\left(-\frac{1}{2}\frac{(y-\mu_y)^2}{\sigma^2_y}\right)\). These expressions are univariate normal distributions: X has mean 渭鈧� and variance 蟽虏鈧�, while Y has mean 渭y and variance 蟽虏y.

Step by step solution

01

Bivariate normal distribution formula

Given that (X, Y) has a bivariate normal distribution, their joint probability density function can be written as: \[ f_{X,Y}(x,y)=\frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} +\frac{(y-\mu_y)^2}{\sigma_y^2} -2\rho \frac{(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}\right] \right) \] where 渭鈧�, 渭y are the means, 蟽虏鈧�, 蟽虏y are the variances and 蟻 is the correlation coefficient between X and Y.
02

Marginal PDF of X

To find the marginal PDF of X, denoted as f鈧�(x), we need to integrate the joint PDF f鈧撯奔(x, y) with respect to y: \[ f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) dy \] Now, let's compute this integration by finding the appropriate factors and exponentials.
03

Computing Marginal PDF of X integration

In order to simplify the integration, we can rewrite the joint PDF f鈧撯奔(x, y) as: \[ f_{X,Y}(x,y) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}} \exp\left(-\frac{1}{2(1-\rho^2)}\left[\frac{(x-\mu_x)^2}{\sigma^2_x}\right]\right) \times \exp\left(-\frac{1}{2(1-\rho^2)}\left[\frac{(y-\mu_y)^2}{\sigma^2_y}-2\rho\frac{(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}\right]\right) \] Notice, now we have two exponentials, and one does not depend on y. We can split the integration accordingly: \[ f_X(x) = \frac{1}{\sqrt{2\pi}\sigma_x} \exp\left(-\frac{1}{2}\frac{(x-\mu_x)^2}{\sigma^2_x}\right) \times \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}\sigma_y\sqrt{1-\rho^2}} \exp\left(-\frac{1}{2(1-\rho^2)}\left[\frac{(y-\mu_y)^2}{\sigma^2_y}-2\rho\frac{(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}\right]\right) dy \]
04

Simplifying the integration for X

Since the first exponential does not depend on y, we can see that the remaining integral is a normal distribution; therefore, its integration will be equal to 1. \[ f_X(x) = \frac{1}{\sqrt{2\pi}\sigma_x} \exp\left(-\frac{1}{2}\frac{(x-\mu_x)^2}{\sigma^2_x}\right) \] This demonstrates that X is normally distributed with mean 渭鈧� and variance 蟽虏鈧�.
05

Marginal PDF of Y

Following the same steps as for finding the marginal PDF of X, we now need to integrate the joint PDF f鈧撯奔(x, y) with respect to x: \[ f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) dx \] By computing and simplifying this integration, we will obtain the univariate normal distribution for Y.
06

Computing and simplifying the integration for Y

Following the same process as in step 3 and step 4, we simplify the integration with respect to x and obtain: \[ f_Y(y) = \frac{1}{\sqrt{2\pi}\sigma_y} \exp\left(-\frac{1}{2}\frac{(y-\mu_y)^2}{\sigma^2_y}\right) \] This demonstrates that Y is normally distributed with mean 渭y and variance 蟽虏y.
07

Conclusion

We have shown that the marginal distributions of X and Y are univariate normal distributions. X is normally distributed with mean 渭鈧� and variance 蟽虏鈧�, and Y is normally distributed with mean 渭y and variance 蟽虏y.

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Key Concepts

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

Marginal Distribution
In the realm of probability and statistics, understanding the marginal distribution is fundamental when dealing with multiple random variables, especially in a joint context like the bivariate normal distribution. Marginal distribution refers to the probability distribution of a single random variable obtained from a joint distribution. When we have a pair of random variables, such as (X, Y), the marginal distribution of one variable, say X, can be found by integrating the joint probability density function over the range of the other variable, Y.
  • For instance, if (X, Y) follows a bivariate normal distribution, the marginal distribution for X can be found by integrating out Y, resulting in the marginal probability density function for X. Similarly, for the marginal distribution of Y, we integrate out X from the joint distribution.
  • Mathematically, if the joint PDF is \(f_{X,Y}(x,y)\), the marginal PDF for X is \(f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) dy\).
  • Likewise, the marginal PDF for Y is given by \(f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) dx\).
The importance of marginal distribution lies in its ability to provide insights into the distribution of an individual variable without the influence of other correlated variables.
Probability Density Function
The Probability Density Function (PDF) is a crucial concept in understanding continuous random variables. It specifies the likelihood of a random variable taking on a particular value. For continuous variables, although the probability at a specific point is zero, the PDF helps calculate the probability of the variable falling within a particular range.
  • The PDF for a continuous random variable Y is usually denoted as \(f_Y(y)\), where \(Y\) can take any value over a continuous range.
  • The PDF must satisfy two primary conditions: it should be non-negative for all possible values of the variable and the integral over its entire range must equal one, representing a total probability of being somewhere within its range.
  • For example, in our bivariate normal distribution scenario, the joint PDF \(f_{X,Y}(x, y)\) incorporates components for both variables, X and Y, highlighting their combined probability distribution.
The PDF is pivotal in fields requiring probability estimation, such as statistics, physics, and engineering, offering a bridge between abstract probability theory and practical data analysis.
Joint Distribution
Joint distribution comes into play when examining the probability distribution of two or more random variables simultaneously. For our case, the joint distribution of the random variables X and Y gives a comprehensive picture of their combined variability and correlation.
  • The expression for the joint PDF of two variables, X and Y, in a bivariate normal distribution is given by:\[f_{X,Y}(x,y)=\frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} +\frac{(y-\mu_y)^2}{\sigma_y^2} -2\rho \frac{(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}\right] \right)\]
  • This equation illustrates how the joint distribution depends not only on the individual distributions of X and Y but also on their correlation, \(\rho\).
  • Understanding the joint distribution is critical when analyzing the dependency between variables and their overall statistical behavior.
Joint distribution is typically explored through functions like the joint PDF for continuous variables or joint probability mass functions for discrete variables, facilitating a deeper understanding of multivariate statistical processes.
Normal Distribution
The Normal Distribution, often referred to as the bell curve or Gaussian distribution, plays a foundational role in statistics and probability theory. It's characterized by its symmetric shape centered around the mean, where data points are more concentrated towards the mean and less frequent as we move away.
  • The normal distribution is defined by two parameters: the mean \(\mu\), which determines the center of the distribution, and the standard deviation \(\sigma\), which controls the spread and width of the curve.
  • In a standard normal distribution, these parameters have specific values where \(\mu = 0\) and \(\sigma = 1\), leading to a unique distribution used for standardization and comparison purposes.
  • For our bivariate case, each individual variable from the joint distribution, like X or Y, showcases a normal distribution once the marginal distribution is derived.
The significance of normal distribution lies in its prevalence in natural and social phenomena, making it a keystone concept for further statistical methods like hypothesis testing and regression analysis.

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