Question

Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(3, \mu_{2}=1, \sigma_{1}^{2}=16, \sigma_{2}^{2}=25\), and \(\rho=\frac{3}{5} .\) Using \(\mathrm{R}\), determine the following probabilities: (a) \(P(3

Short answer

The solutions are the return results of the R scripts as discussed in each step.

Step-by-step solution

Identify Parameters

Identify the mean (\(\mu\)), variance (\(\sigma^2\)), and correlation coefficient (\(\rho\)) for the bivariate normal distribution. Here, \(\mu_{1} = 3, \mu_{2} = 1, \sigma_{1}^{2} = 16, \sigma_{2}^{2} = 25\), and \(\rho = 3/5\).

Compute Probability (a)

Find the probability \(P(3 < Y < 8)\). Using R, compute this as \(``pnorm(8, mean=1, sd=sqrt(25))-pnorm(3, mean=1, sd=sqrt(25))``\).

Compute Probability (b)

Find the probability \(P(3 < Y < 8 | X = 7)\). First calculate the conditional mean: \(\mu_{2|1} = \mu_{2} + \rho(\sigma_{2}/\sigma_{1})*(7-\mu_{1})\) and conditional variance: \(\sigma_{2|1}^{2} = \sigma_{2}^{2}(1-\rho^{2})\). Then compute the probability using R as \(``pnorm(8, mean=mu_2_1, sd=sqrt(sigma_2_1^2))-pnorm(3, mean=mu_2_1, sd=sqrt(sigma_2_1^2))``\).

Compute Probability (c)

Find the probability \(P(-3 < X < 3)\). Compute this in R as \(``pnorm(3, mean=3, sd=sqrt(16))-pnorm(-3, mean=3, sd=sqrt(16))``\).

Compute Probability (d)

Find the probability \(P(-3 < X < 3 | Y = -4)\). Calculate the conditional mean: \(\mu_{1|2} = \mu_{1} + \rho(\sigma_{1}/\sigma_{2})*(-4-\mu_{2})\) and conditional variance: \(\sigma_{1|2}^{2} = \sigma_{1}^{2}(1-\rho^{2})\). Compute this in R as \(``pnorm(3, mean=mu_1_2, sd=sqrt(sigma_1_2^2))-pnorm(-3, mean=mu_1_2, sd=sqrt(sigma_1_2^2))``\).

Key concepts

Conditional Probability

Understanding the concept of conditional probability is essential in statistics, as it allows us to calculate the likelihood of an event occurring given the occurrence of another event.

Conditional probability is defined by the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
where \( P(A|B) \) is the conditional probability of event \( A \) occurring given that event \( B \) has occurred, \( P(A \cap B) \) is the joint probability of both events occurring, and \( P(B) \) is the probability of event \( B \).

In the case of the bivariate normal distribution, we often want to know something like the probability of \( Y \) being in a certain range, given a specific value of \( X \). For instance, if \( X \) is the height of a person and \( Y \) is their weight, we might want to know the range of possible weights for a person of a specific height. This is where conditional probability plays a critical role.

R Statistical Software

R is a powerful tool for statistical analysis and graphical representation. It is widely used by statisticians and data scientists for its extensive library of statistical and graphical methods.

When dealing with complex probability distributions, like the bivariate normal distribution, R can be invaluable. With built-in functions like pnorm for the normal distribution, R allows us to calculate probabilities and other statistical parameters rapidly. The use of R in solving the given textbook exercises provides a practical example of how technology can aid in complex statistical calculations. By supplying commands to calculate the conditional probabilities, R helps students and researchers focus more on understanding the concepts rather than getting bogged down in the computational details.

Statistical Parameters

In any probability distribution, statistical parameters such as mean \( (\mu) \), variance \( (\sigma^2) \), and correlation coefficient \( (\rho) \) play a pivotal role in describing the characteristics of the distribution.

Mean is the average value showing where data points tend to cluster; in the bivariate distribution, each variable has its own mean. Variance measures the spread or variability of the distribution, with each variable again having its individual variance. Lastly, the correlation coefficient quantifies the strength and direction of the relationship between the two variables involved in a bivariate distribution.

Understanding these parameters is crucial because they influence the shape and properties of the probability distribution. They provide insight into the nature of the randomness we are dealing with and are the foundational elements for calculating probabilities.

Probability Distribution

A probability distribution is a mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.

The bivariate normal distribution is a type of probability distribution that describes the joint variability of two continuous random variables. It is an extension of the normal (or Gaussian) distribution. In a bivariate normal distribution, knowing the value of one variable can give us information about the probabilities for the other variable, and this is where understanding conditional probabilities within the context of this distribution becomes pertinent.

The bivariate normal distribution is characterized by bell-shaped curves and depends heavily on its mean, variance, and correlation coefficient. These attributes influence its overall shape and the way the variables are related, greatly assisting in predictive analytics.