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 exact solution for \(\rho\) requires numerical methods or iterative approximation methods which are usually done with the aid of software tools or calculators that have statistical functionalities. Thus, we don't provide a numerical answer.

Step-by-step solution

Calculate Z scores for the given range

First, compute the Z scores corresponding to the Y values 4 and 16, using the formula \(Z=\frac{Y-\mu_{2}}{\sigma_{2}}\). For Y=4, the Z value is \(-1.2\) and for Y=16, the Z value is \(1.2\) because \(\mu_{2}=10\) and \(\sigma_{2}=5\)

Calculate the probability using the Z table

Next, use the standard normal distribution table to find the probabilities corresponding to the Z scores of -1.2 and 1.2. The area under the standard normal curve from \(-\infty\) to \(Z=1.2\) is approximately 0.8849 and from \(-\infty\) to \(Z=-1.2\) is approximately 0.1151. You find the probability between these Z values by subtracting the two probabilities, leading to \(0.8849 - 0.1151 = 0.7698\)

Use the given conditional probability to find \(\rho\)

We know that the probability between these two Z values is supposed to be 0.954 according to the problem. This clearly isn't matching with the calculated probability 0.7698. The main reason that the calculated Z scores aren't giving the correct probability is because \(\rho\) may not be 0. It is given that \(\rho > 0\), so the dependence between X and Y must be considered. You must use the given conditional probability to determine \(\rho\). The solution won't be a straightforward calculation and would require numerical methods or iterative approximation methods to find the exact value of \(\rho\) that will give the desired conditional probability of \(0.954\).

Determine \(\rho\) using numerical methods

This step will usually involve computer software or calculators that have numerical or statistical functionalities. These would try different values of \(\rho\) and see when the calculated probability matches the given probability. Since we cannot illustrate this step without using a specific software or calculator, the exact value of \(\rho\) is not determined here.