Question

Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(\mu_{2}=0, \sigma_{1}^{2}=\sigma_{2}^{2}=1\), and correlation coefficient \(\rho .\) Find the distribution of the random variable \(Z=a X+b Y\) in which \(a\) and \(b\) are nonzero constants.

Short answer

The random variable \(Z\) = \(aX+bY\) is normally distributed with parameters \(N(0, a^2+b^2+2ab\rho)\).

Step-by-step solution

Define the random variable \(Z\)

First, let's define the random variable \(Z\). We are given \(Z=aX+bY\). This implicates that \(Z\) is a normal random variable because a linear combination of normal random variables is also normal (if \(X\) and \(Y\) are jointly normal). The challenge is to find the parameters of this normal distribution namely its mean and variance.

Find the expected value and variance of \(Z\)

From the properties of expectation and variance, the expected value of \(Z\) can be expressed as \[E[Z] = E[aX+bY] = aE[X] + bE[Y] = a\mu_1 + b\mu_2 = 0.\] The variance of \(Z\) can be obtained as: \[Var(Z) = Var(aX+bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y),\] where \(Cov(X,Y)\) is given by \(\rho\sigma_1\sigma_2\). Substituting the given values, we will have \(Var(Z) = a^2 + b^2 + 2ab\rho\).

Express the distribution of \(Z\)

Now we know that \(Z\) is a normal random variable with mean 0 and variance \(a^2 + b^2 + 2ab\rho\), we can say \(Z\) follows a normal distribution \(N(0, a^2 + b^2 + 2ab\rho)\). This is the distribution of the random variable \(Z\)=\(aX+bY\).

Key concepts

Linear Combination of Random Variables

In the context of probability and statistics, the term linear combination of random variables refers to an expression constructed from a set of random variables by multiplying each with a constant and then adding the results. This operation is fundamental to understanding many processes in statistics, such as regression analysis, and in the study we're looking at, the linear combination defines the new random variable, denoted as \( Z = aX + bY \).

One important property to remember is that if random variables \(X\) and \(Y\) are separately normal and are part of a bivariate normal distribution, as stated in the exercise, the linear combination \(Z\) will also follow a normal distribution. These relationships make the analysis and prediction of complex variables possible, facilitating the understanding of diverse phenomena in various scientific fields, from finance to engineering.

Expected Value and Variance

Understanding expected value and variance is critical in the realm of probability theory as they measure the central tendency and the dispersion of a probability distribution, respectively. The expected value, or the mean, of a random variable provides a measure of the 'center' of its distribution. In simpler terms, it tells us the average outcome we can expect if we were to repeat an experiment many times.

The variance, on the other hand, gives us an idea of how spread out the values of the random variable are around the mean. In terms of calculations, for a given linear combination of variables \(Z = aX + bY\), the expected value is \(E[Z] = a\text{E}[X] + b\text{E}[Y]\), and the variance is a bit more complex: \(Var(Z) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)\), incorporating the idea of covariance, which measures how much two random variables change together. A clear and practical understanding of these concepts is vital for students and professionals alike to correctly interpret data and make informed decisions.

Normal Random Variable

The normal random variable is a cornerstone in the study of probability and statistics due to its prevalence in the natural and social sciences. A normal random variable has a distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean.

When a random variable is described as normal, it follows the well-known bell-shaped curve, characterized entirely by its mean \( \text{E}[X] \) and its variance \( Var(X) \). The centrality of the normal distribution can be attributed to the Central Limit Theorem, which tells us that under certain conditions, the sum (or average) of a large number of independent random variables will be approximately normal, regardless of their underlying distributions. This property of normal random variables makes them invaluable for statistical modeling and hypothesis testing where predictions and inferences about populations from samples are made.