For positive and perfectly collated, random variables. One of the regression coefficient is 1.3 and the standard deviation of X is two then the variation of Y is.

To find the variance of \( Y \) when given that one of the regression coefficients (likely \( \beta_1 \)) is 1.3 and the standard deviation of \( X \) is 2, we can use the following relationship:

\[
\beta_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}
\]

Given:
- Regression coefficient \( \beta_1 = 1.3 \)
- Standard deviation of \( X \), \( \sigma_X = 2 \) (so \( \text{Var}(X) = \sigma_X^2 = 4 \))

First, we calculate the covariance \( \text{Cov}(X, Y) \) using the formula for the regression coefficient:

\[
\text{Cov}(X, Y) = \beta_1 \times \text{Var}(X)
\]

\[
\text{Cov}(X, Y) = 1.3 \times 4 = 5.2
\]

The variance of \( Y \) can be found using the formula:

\[
\text{Var}(Y) = \beta_1^2 \times \text{Var}(X) + \text{Var}(\epsilon)
\]

However, the problem does not mention the variance of the error term (\( \text{Var}(\epsilon) \)). If we assume that there is no error or that the error term has a variance of zero, then:

\[
\text{Var}(Y) = \beta_1^2 \times \text{Var}(X)
\]

\[
\text{Var}(Y) = (1.3)^2 \times 4 = 1.69 \times 4 = 6.76
\]

Thus, the variance of \( Y \) is \( \text{Var}(Y) = 6.76 \).

### Do you want more details or have any questions?

### Related Questions:
1. How would the variance of \( Y \) change if the variance of the error term is not zero?
2. What is the covariance between \( X \) and \( Y \) if the regression coefficient is different?
3. How does changing the standard deviation of \( X \) affect the variance of \( Y \)?
4. What assumptions are necessary for the linear regression model to hold?
5. How would the interpretation change if \( \beta_1 \) was negative instead of positive?

**Tip:** When dealing with regression analysis, understanding the role of variance and covariance is crucial, as they are foundational to determining how changes in one variable affect another.

Learn how to calculate the variance of Y when given a regression coefficient of 1.3 and a standard deviation of X equal to 2. This problem explores key concepts in regression analysis, including the relationship between variance, covariance, and the regression coefficient. Suitable for undergraduate-level statistics courses.

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