If the correlation coefficient between the random variables X and Y be negative, then which of the following statements is true or false
`E(XY)ltE(X)E(Y)`

Answer

To determine whether the statement \( E(XY) < E(X)E(Y) \) is true or false given that the correlation coefficient between the random variables \( X \) and \( Y \) is negative, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Correlation Coefficient**:
The correlation coefficient \( r \) between two random variables \( X \) and \( Y \) is defined as:
\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \]
where \( \text{Cov}(X, Y) \) is the covariance between \( X \) and \( Y \), and \( \sigma_X \) and \( \sigma_Y \) are the standard deviations of \( X \) and \( Y \), respectively.

2. **Express Covariance**:
The covariance can be expressed in terms of expectations:
\[ \text{Cov}(X, Y) = E(XY) - E(X)E(Y) \]

3. **Substitute into the Correlation Formula**:
Substituting the expression for covariance into the correlation coefficient formula gives:
\[ r = \frac{E(XY) - E(X)E(Y)}{\sigma_X \sigma_Y} \]

4. **Analyze the Condition for Negative Correlation**:
For the correlation coefficient \( r \) to be negative, the numerator \( E(XY) - E(X)E(Y) \) must be negative. This leads to:
\[ E(XY) - E(X)E(Y) < 0 \]
Rearranging this inequality gives:
\[ E(XY) < E(X)E(Y) \]

5. **Conclusion**:
Since we have established that \( E(XY) < E(X)E(Y) \) when the correlation coefficient \( r \) is negative, we can conclude that the statement \( E(XY) < E(X)E(Y) \) is **true**.

### Final Answer:
The statement \( E(XY) < E(X)E(Y) \) is **True**.

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