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Magazine Chapter 5: Problem 90
$$ \begin{aligned} \text { If } f_{X Y}(x, y)=\frac{1}{1.2 \pi} \exp \left\\{\frac{-1}{0.72}\left[(x-1)^{2}\right.\right.\\\ &\left.\left.-1.6(x-1)(y-2)+(y-2)^{2}\right]\right\\} \end{aligned} $$ determine \(E(X), E(Y), V(X), V(Y),\) and \(\rho\) by reorganizing the parameters in the joint probability density function.
Short Answer
Expert verified
\(E(X) = 1, E(Y) = 2, V(X) = 0.6, V(Y) = 0.6, \rho = 0.8\).
Step by step solution
01
Identify the Joint PDF form
The given joint PDF can be identified as a bivariate normal distribution. It follows the form of a multivariate normal distribution due to its exponential expression. This form is typically written with a quadratic form in the exponent, which matches the structure provided in the problem.
02
Rearrange the Exponent Quadratic Form
The quadratic expression is given as \(-\frac{1}{0.72}((x-1)^{2} - 1.6(x-1)(y-2) + (y-2)^{2})\). Rearrange this to match the form \(\frac{1}{2(1-\rho^2)}(\frac{(x-\mu_x)^2}{\sigma_x^2} - 2\rho\frac{(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y} + \frac{(y-\mu_y)^2}{\sigma_y^2})\).
03
Extract Mean from the Parameters
In the standard form, the terms \(x-1\) and \(y-2\) indicate that the means are \(\mu_x = 1\) and \(\mu_y = 2\), respectively.
04
Determine the Variances and Correlation Coefficient
By comparing the rearranged form to the standard form, identify the coefficients of the quadratic terms and correlation coefficient term. Given the factor 0.72 in the denominator, deduce the variances and \(\rho\). The variance terms are extracted to be \(\sigma_x^2 = \sigma_y^2 = 0.6\) and the correlation coefficient \(\rho = 0.8\), satisfying the factor inside the exponential.
05
Conclude Expected Values, Variances, and Correlation
Using the extracted values, conclude that \(E(X) = 1\), \(E(Y) = 2\). The variances are \(V(X) = 0.6\), \(V(Y) = 0.6\), and the correlation coefficient \(\rho = 0.8\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Joint Probability Density Function
In statistics, the bivariate normal distribution is a common way to describe the probability of two continuous random variables happening simultaneously. This is represented by the {Joint Probability Density Function (PDF)}. The provided function describes the likelihood that a pair of values \(X\) and \(Y\) appear together.
For any pair of values \(x\) and \(y\), this function gives the height of the joint PDF surface at that point. This can help in determining how often you can expect to see particular pairs of values. The exponential part of the function tends to form a bell-shaped surface, characteristic of normal distributions, over the plane defined by \(x\) and \(y\).
The specific form presented here, shown as an exponential of a quadratic term, is a typical representation of bivariate normal distribution PDF. It's written with parameters that reflect means, variances, and the correlation coefficient (more on this later).Understanding this concept is crucial, as it stands as the backbone for predicting outcomes for two related continuous random variables. Knowing how to use and interpret the joint PDF enables you to calculate the likelihoods of different scenarios effectively.
For any pair of values \(x\) and \(y\), this function gives the height of the joint PDF surface at that point. This can help in determining how often you can expect to see particular pairs of values. The exponential part of the function tends to form a bell-shaped surface, characteristic of normal distributions, over the plane defined by \(x\) and \(y\).
The specific form presented here, shown as an exponential of a quadratic term, is a typical representation of bivariate normal distribution PDF. It's written with parameters that reflect means, variances, and the correlation coefficient (more on this later).Understanding this concept is crucial, as it stands as the backbone for predicting outcomes for two related continuous random variables. Knowing how to use and interpret the joint PDF enables you to calculate the likelihoods of different scenarios effectively.
Expected Values
The expected value (often symbolized as \(E(X)\) or \(E(Y)\)) of a random variable is its long-term average, or the mean value you would expect if you were to observe many instances of that variable. It provides a core reference point around which data tends to cluster. Calculating expected values involves summing each possible outcome, weighted by its probability.
For bivariate normal distributions, the expected values of the variables \(X\) and \(Y\) indicate their respective central tendencies. In the given function, these expected values can be determined directly from the parameters inside the quadratic expression. Specifically, when the mean adjustments \((x-1)\) and \((y-2)\) appear in the expression, they indicate that the expected values for \(X\) and \(Y\) are 1 and 2, respectively.
These expected values play a vital role in understanding the distribution's shape, primarily identifying the distribution鈥檚 center. By always considering expected values, you are better prepared to predict and gauge the normal tendencies of related variables.
For bivariate normal distributions, the expected values of the variables \(X\) and \(Y\) indicate their respective central tendencies. In the given function, these expected values can be determined directly from the parameters inside the quadratic expression. Specifically, when the mean adjustments \((x-1)\) and \((y-2)\) appear in the expression, they indicate that the expected values for \(X\) and \(Y\) are 1 and 2, respectively.
These expected values play a vital role in understanding the distribution's shape, primarily identifying the distribution鈥檚 center. By always considering expected values, you are better prepared to predict and gauge the normal tendencies of related variables.
Variances
Variance measures the amount of spread or variability in the values of a random variable. For a bivariate normal distribution, the variances \(V(X)\) and \(V(Y)\) indicate how much each variable spreads out around its mean.They are shown in the denominator of the terms of the expression outlining the bivariate function.
In this scenario, the quadratic terms are also part of the variance calculation. Recognizing the proper denomination and aligning it with the standard form reveals that the variances for both \(X\) and \(Y\) are 0.6.
These variances are particularly important for understanding how broad or narrow the distribution is around its mean. More significant variance indicates a wider spread of probable values, while smaller values show the outcomes are more consistently clustered around the expected value.
In this scenario, the quadratic terms are also part of the variance calculation. Recognizing the proper denomination and aligning it with the standard form reveals that the variances for both \(X\) and \(Y\) are 0.6.
These variances are particularly important for understanding how broad or narrow the distribution is around its mean. More significant variance indicates a wider spread of probable values, while smaller values show the outcomes are more consistently clustered around the expected value.
Correlation Coefficient
The correlation coefficient, denoted by \(\rho\), reflects the degree of linear relationship between the two random variables \(X\) and \(Y\). In a bivariate normal distribution, \(\rho\) can vary from -1 to +1. A value of +1 indicates a perfect positive linear relationship, while -1 indicates a perfect negative linear relationship. A value of 0 suggests no linear correlation.
This characteristic is crucial as it defines how changes in one variable might predict changes in another. In the given problem, the correlation coefficient is noted to be 0.8, showing a substantial positive relationship. This signals that as \(X\) increases, \(Y\) is also likely to increase, and vice versa.Identifying \(\rho\) involves examining the coefficient of the cross term \((x-1)(y-2)\) within the PDF's quadratic form. Manipulating the PDF into the standard form, \(\rho\) emerges from matching the specific framework coefficients.Understanding the correlation is essential for making predictions based on these interconnected variables. Its real-world applications are numerous, from finance to natural sciences, enhancing the reliability of any conclusions drawn from statistical analyses.
This characteristic is crucial as it defines how changes in one variable might predict changes in another. In the given problem, the correlation coefficient is noted to be 0.8, showing a substantial positive relationship. This signals that as \(X\) increases, \(Y\) is also likely to increase, and vice versa.Identifying \(\rho\) involves examining the coefficient of the cross term \((x-1)(y-2)\) within the PDF's quadratic form. Manipulating the PDF into the standard form, \(\rho\) emerges from matching the specific framework coefficients.Understanding the correlation is essential for making predictions based on these interconnected variables. Its real-world applications are numerous, from finance to natural sciences, enhancing the reliability of any conclusions drawn from statistical analyses.
