Problem 74 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that the random variables \(X\) and \(Y\) have a bivariate normal pdf with \(\mu_{X}=56, \mu_{Y}=11, \sigma_{X}^{2}=1.2\), \(\sigma_{Y}^{2}=2.6\), and \(\rho=0.6\). Compute \(P(10

Step-by-Step Solution

Answer

The calculated probabilities for the events \(P(10

Step 1: Calculate Conditional Mean and Variance

First, evaluate the conditional mean and variance of \(Y\) given \(X\ =\ 55\). Use the following formulas: The conditional mean, \(E(Y\ |\ X\ =\ x) = \mu_{Y}+\rho \frac{\sigma_{Y}}{\sigma_{X}}(x-\mu_{X})\). The conditional variance, \(\text{Var}(Y\ |\ X\ =\ x) = \sigma_{Y}^{2}(1-\rho^{2})\).

Step 2: Compute Probability for the First Part

Use the calculated conditional mean and variance in the standard normal probability to calculate the required probability \(P(10

Step 3: Compute Probability for the Second Part

The second part involves finding the mean \(\bar{Y}\) for four observations. Given \(n=4\), the standard error \(\frac{\sigma}{\sqrt{n}}\) should be used, and this will become the new standard deviation for \(\bar{Y}\). The process is similar to the one in step 2. Calculate the \(Z\) scores for 10.5 and 11, find their corresponding probabilities, and subtract \(\Phi(Z_1)\) from \(\Phi(Z_2)\) to find the required probability.

Key Concepts

Conditional Probability

Understanding conditional probability is akin to figuring out the likelihood of an event occurring, provided another event has already transpired. This concept is integral when dealing with bivariate distributions, like the bivariate normal distribution in our exercise.

Take for instance, the question asking for the probability that random variable Y falls between 10 and 10.5 given that X is fixed at 55. This scenario requires you to consider only those outcomes of Y that are associated with the given value of X. Conditional probability essentially filters the sample space, considering only a subset where certain conditions are met.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution often symbolized as Z. It has a mean of 0 and a standard deviation of 1. Any normal distribution can be converted into the standard normal distribution using the Z-Score formula. This transformation is known as standardization.

In our solved problem, the transformation to the standard normal distribution is crucial for calculating the required probabilities, as standard normal tables or software can then be used to find the associated probabilities easily.

Z-Score

The Z-Score serves as a bridge between an individual data point in a normal distribution and the standard normal distribution. It's a measure of how many standard deviations an element is from the mean.

The formula is given by: \[ Z = \frac{X - \mu}{\sigma} \]
Where X is the data point, \mu is the mean, and \sigma is the standard deviation. This standardization process enables comparability across different normal distributions and simplifies the calculation of probabilities.

Variance and Standard Deviation

Variance measures the spread or dispersion of a set of data points within a data set, indicating how far the points lie from the mean. Standard deviation is the square root of variance and provides a clear measure of spread in the same units as the data.

In our context, the conditional variance is given by the formula: \[ \text{Var}(Y \ | \ X = x) = \sigma_{Y}^{2}(1-\rho^{2}) \]
Here, \rho represents the correlation coefficient between X and Y. A crucial step for solving part one of the exercise is calculating the variance of Y given X, which impacts the width of the conditional distribution of Y.

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Problem 75

Other exercises in this chapter

Problem 72

Suppose that \(X\) and \(Y\) have a bivariate normal distribution with \(\operatorname{Var}(X)=\operatorname{Var}(Y)\). (a) Show that \(X\) and \(Y-\rho X\) are

View solution

Problem 73

Suppose that \(X\) and \(Y\) have a bivariate normal distribution. (a) Prove that \(X+Y\) has a normal distribution when \(X\) and \(Y\) are standard normal ran

View solution

Problem 75

If the joint pdf of the random variables \(X\) and \(Y\) is $$ f_{X, Y}(x, y)=k e^{-(2 / 3)\left[(1 / 4) x^{2}-(1 / 2) x y+y^{2}\right]} $$ find \(E(X), E(Y), \

View solution

Problem 76

Give conditions on \(a>0, b>0\), and \(u\) so that $$ f_{X, Y}(x, y)=k e^{-\left(a x^{2}-2 u x y+b y^{2}\right)} $$ is the bivariate normal density of random va

View solution

Key Concepts

Conditional Probability

Understanding conditional probability is akin to figuring out the likelihood of an event occurring, provided another event has already transpired. This concept is integral when dealing with bivariate distributions, like the bivariate normal distribution in our exercise.

Take for instance, the question asking for the probability that random variable Y falls between 10 and 10.5 given that X is fixed at 55. This scenario requires you to consider only those outcomes of Y that are associated with the given value of X. Conditional probability essentially filters the sample space, considering only a subset where certain conditions are met.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution often symbolized as Z. It has a mean of 0 and a standard deviation of 1. Any normal distribution can be converted into the standard normal distribution using the Z-Score formula. This transformation is known as standardization.

In our solved problem, the transformation to the standard normal distribution is crucial for calculating the required probabilities, as standard normal tables or software can then be used to find the associated probabilities easily.

Z-Score

The Z-Score serves as a bridge between an individual data point in a normal distribution and the standard normal distribution. It's a measure of how many standard deviations an element is from the mean.

The formula is given by: \[ Z = \frac{X - \mu}{\sigma} \]
Where X is the data point, \mu is the mean, and \sigma is the standard deviation. This standardization process enables comparability across different normal distributions and simplifies the calculation of probabilities.

Variance and Standard Deviation

Variance measures the spread or dispersion of a set of data points within a data set, indicating how far the points lie from the mean. Standard deviation is the square root of variance and provides a clear measure of spread in the same units as the data.

In our context, the conditional variance is given by the formula: \[ \text{Var}(Y \ | \ X = x) = \sigma_{Y}^{2}(1-\rho^{2}) \]
Here, \rho represents the correlation coefficient between X and Y. A crucial step for solving part one of the exercise is calculating the variance of Y given X, which impacts the width of the conditional distribution of Y.