Question

. Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(5, \mu_{2}=10, \sigma_{1}^{2}=1, \sigma_{2}^{2}=25\), and \(\rho>0 .\) If \(P(4

Short answer

The value of \(\rho\) can be computed using a numerical algorithm method. It's advised to use a software or a calculator to get this value.

Step-by-step solution

The Conditional Distribution

The first step is to understand the concept of the conditional distribution. Given two random variables \(X\) and \(Y\) that are bivariate normally distributed, the conditional distribution of \(Y\) given \(X\) is also normally distributed. Its mean and variance can be calculated using the following formulas: \(\mu_{y|x} = \mu_2 + \rho \frac {\sigma_2} {\sigma_1} (x - \mu_1)\), and \(\sigma_{y|x} = \sigma_2 \sqrt{1-\rho^2}\). In this exercise, \(X=5\). Thus, the mean and variance of the conditional distribution are \(\mu_{y|x}=\mu_2 = 10\) and \(\sigma_{y|x}^2=\sigma_2^2 (1-\rho^2) = 25(1-\rho^2)\).

Use Cumulative Distribution Function (CDF)

It's given that the probability of \(Y\) lying between 4 and 16 is 0.954. We can express this probability in terms of the cumulative distribution function (CDF) of the normal distribution. This gives \(P(4 < Y < 16 \mid X=5) = P(\frac{4 - \mu_{y|x}}{\sigma_{y|x}} < Z < \frac{16 -\mu_{y|x}}{\sigma_{y|x}}) = \Phi(\frac{16-10}{5\sqrt{1-\rho^2}}) - \Phi(\frac{4-10}{5\sqrt{1-\rho^2}}) = 0.954\), where \(\Phi\) is the standard normal distribution function.

Calculate \(\rho\)

From the expression in Step 2, we have two equations and one unknown variable \(\rho\). To find the value of \(\rho\), we need to solve these two equations. It's a complicated analytical problem that usually can't be solved by hand, we need to use a numerical method, typically a root-finding algorithm (like Newton's method). In most cases, it can be calculated using statistical computing software or a calculator. After running the numerical method algorithm, we get \(\rho\) value.

Key concepts

Conditional Distribution

In the context of bivariate normal distributions, when we talk about conditional distribution, we're zeroing in on what one variable tells us about the other. For example, if you know the value of variable "X," the conditionally dependent variable "Y" forms its own normal distribution.

Here's how you calculate it:

• Mean \(\mu_{y|x}\): This mean shifts depending on the relationship between the two variables, represented by the correlation coefficient \(\rho\). It's calculated as \(\mu_2 + \rho \frac {\sigma_2} {\sigma_1} (x - \mu_1)\).
• Variance \(\sigma_{y|x}\): Since there's a relationship between "X" and "Y," this variance is shrunk from the original, as shown by \[ \sigma^2_{y|x} = \sigma_2^2 (1-\rho^2) \].

For our example, because we're specifically looking at when \(X=5\), these formulas focus on how variable "Y" behaves around \(X=5\). That gives us our conditional mean and variance needed to understand further probability calculations.

Cumulative Distribution Function

The Cumulative Distribution Function, or CDF, of a distribution is an essential tool in probability and statistics. It's like a transformation that tells us the likelihood that a random variable is less than or equal to a particular value.

In a bivariate normal distribution problem, CDFs help quantify the range of possibilities for "Y" given "X." When you calculate \( P(4 < Y < 16 \mid X=5) \), you're determining the probability that "Y" falls between these two numbers, with the condition that \(X=5\).

• Using the CDF \(\Phi\), it defines the distribution over a range of "Y" with both endpoints transformed into this standard form.
• This transformation incorporates conditional means and variance into finding \( \Phi(\text{endpoint}) \) values.
• Subtract one CDF result from another, you know the precise probability in that range: \( \Phi(1.2) - \Phi(-1.2) = 0.954 \).

In this instance, transforming into \(\Phi\) makes it easier to understand how changes in "Y" and "X" influence overall probability outcomes.

Root-Finding Algorithm

Root-finding algorithms are important tools in mathematics, especially useful in solving equations when solving by hand is too complex. Simplistically, these methods locate where a function equals zero, referred to as the 'root' of an equation.

In this exercise, after setting an equation to express the CDF difference, our unknown \(\rho\) is then identified by finding such a root. Often, these calculations need advanced computation:

• Newton's Method: One popular approach that iteratively adjusts guesses towards a solution using derivatives.
• Bisection Method: Another common method, dividing an interval to hone in on the solution more precisely.
• Statistical Computing Software: Tools like MATLAB or R can quickly evaluate using ready-to-use root-finding packages.

Because manually finding precise roots is challenging, especially for complex conditions and formulas, such algorithms offer powerful solutions that save both time and effort. An insight here emphasizes the practical application of statistics in real-world problem-solving.