The systolic and diastolic blood pressure values (mm Hg) are...

The systolic and diastolic blood pressure values (mm Hg) are the pressures when the heart muscle contracts and relaxes (denoted as $Y$ and $X,$ respectively). Over a collection of individuals, the distribution of diastolic pressure is normal with mean 73 and standard deviation $8 .$ The systolic pressure is conditionally normally distributed with mean $1.6 x$ when $X=x$ and standard deviation of $10 .$ Determine the following: (a) Conditional probability density function $f_{Y \mid 73}(y)$ of $Y$ given $X=73$ (b) $P(Y<115 \mid X=73)$ (c) $E(Y \mid X=73)$ (d) Recognize the distribution $f_{X Y}(x, y)$ and identify the mean and variance of $Y$ and the correlation between $X$ and $Y$

The systolic and diastolic blood pressure values (mm Hg) are the pressures when the heart muscle contracts and relaxes (denoted as $Y$ and $X,$ respectively). Over a collection of individuals, the distribution of diastolic pressure is normal with mean 73 and standard deviation $8 .$ The systolic pressure is conditionally normally distributed with mean $1.6 x$ when $X=x$ and standard deviation of $10 .$ Determine the following:
(a) Conditional probability density function $f_{Y \mid 73}(y)$ of $Y$ given $X=73$
(b) $P(Y<115 \mid X=73)$
(c) $E(Y \mid X=73)$
(d) Recognize the distribution $f_{X Y}(x, y)$ and identify the mean and variance of $Y$ and the correlation between $X$ and $Y$

Key Concepts

Conditional Probability Density Function

This concept refers to the probability density of a random variable given that another variable is fixed at a specific value. It is derived from the joint distribution of the two variables and provides the distribution of outcomes for the variable of interest subject to the known value of the other variable.

Bivariate Normal Distribution

A bivariate normal distribution is a two?variable joint distribution where each variable is normally distributed and any linear combination of the two is also normally distributed. This model is characterized by the means, variances of each variable, and the covariance or correlation between the variables, and it simplifies analysis of conditional distributions and related probabilities.

Conditional Expectation

Conditional expectation is the expected value (mean) of one random variable given a specific value of another. It is a fundamental concept used to understand how the mean of one variable varies in relation to another and serves as a key parameter in conditional distributions and linear regression models.

Law of Total Variance

The law of total variance decomposes the overall variance of a random variable into the expected value of the conditional variances plus the variance of the conditional expectations. This decomposition is crucial when analyzing joint distributions since it links the spread of the variable’s distribution to the uncertainty remaining after conditioning on another variable.

Correlation

Correlation is a measure of the linear relationship between two random variables. It is defined as the covariance of the variables normalized by the product of their standard deviations. In joint distributions, the correlation statistic quantifies the strength and direction of the association between the two variables.

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