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Chapter 3: Problem 5

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

Expert verified
The correlation coefficient \(\rho = 1.31\), however note that \(\rho\) ranges from -1 to 1. So there must be some misunderstanding, it's necessary to revisit the problem presented.

Step by step solution

01

Calculate z-scores for given Y limits

First, convert the given Y-values (4 & 16) to z-scores using the formula \(Z=\frac{{(X-\mu)}}{{\sigma}}\). Here X denotes the Y-values (4, 16), 碌 is the mean \(\mu_{2}=10\), and 蟽 is the standard deviation \(\sigma_{2}= \sqrt{25} = 5\). Therefore, z-scores would be \(Z_1= \frac{{(4 - 10)}}{{5}} = -1.2\) and \(Z_2= \frac{{(16 - 10)}}{{5}} = 1.2\)
02

Calculate X's Z-score

Since X=5 and given the parameters, calculate the Z score for X using the same formula as above. Here X is 5, 碌 is the mean \(\mu_{1}=5\), and 蟽 is the standard deviation \(\sigma_{1}= \sqrt{1} = 1\). Therefore, Z for X will be \(Z_x= \frac{{(5 - 5)}}{{1}} = 0\)
03

Find value from standard normal distribution table

Recall that P(4<Y<16 | X=5)=0.954 translates to a Z-score interval between -1.2 and 1.2. Find the values corresponding to these Z-scores in the standard normal distribution table. The values are: \(F(1.2) - F(-1.2) = 0.8849 - 0.1151 = 0.7698\)
04

Solve for rho

Upon knowing that the cumulative density value from the standard normal distribution table is 0.7698 we equate it to the given value of 0.954 to solve for rho, we use \(\rho = \frac{{0.954 - 0.5}}{{0.7698 - 0.5}}\)

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