Question

Let \(X \sim N(\mu X, \sigma _X ^2)\) and \(Y \sim N(\mu _y, \sigma _y ^2)\) Which of the following is/are NOT correct?

• A. The area \(F(X)= \frac1{\sigma _x \sqrt{2\pi}}\)\(\int_{-\infty}^{\mu_x} e^{-\frac1{2}(\frac{X-\mu_x}{\sigma_x})^2} \)dx is 1.
• B. The areas under the normal probability curve between the ordinates at \(\mu_x ± 3\sigma_x\) , and \(\mu_y ± 3\sigma_y\) are 0.9544 and 0.9973, respectively.
• C. For variable X,
  Quartile Deviation: Mean Absolute Deviation: Standard Deviation ≅ \(\frac2{3} \sigma_x : \frac4{5}\sigma_x : \sigma_x\)
• D. If X and Y are independent, then \((X-Y) \sim N(\mu_x - \mu_y,  σ^2_x +σ^2_y)\).

Answer 1

The Correct Option is A, B

To determine which statements are not correct regarding normal distribution and its properties, let us analyze each given option one by one.

1. Statement 1: The area \(F(X)= \frac1{\sigma_x \sqrt{2\pi}}\)\(\int_{-\infty}^{\mu_x} e^{-\frac1{2}(\frac{X-\mu_x}{\sigma_x})^2} \)dx is 1.

   Explanation: In a normal distribution, the total area under the probability density function from \(-\infty\) to \(+\infty\) is equal to 1. However, this statement refers to integrating from \(-\infty\) to \(\mu_x\), which gives an area of 0.5 for a symmetric distribution like the normal distribution. Therefore, this statement is incorrect.

2. Statement 2: The areas under the normal probability curve between the ordinates at \(\mu_x ± 3\sigma_x\) and \(\mu_y ± 3\sigma_y\) are 0.9544 and 0.9973, respectively.

   Explanation: The area within \(\pm 3\sigma\) in a normal distribution is approximately 0.9973, not 0.9544. For \(\pm 2\sigma\), the area is approximately 0.9545. Therefore, the statement regarding the area as 0.9544 is incorrect.

3. Statement 3: For variable X,
   Quartile Deviation: Mean Absolute Deviation: Standard Deviation ≅ \(\frac2{3} \sigma_x : \frac4{5}\sigma_x : \sigma_x\)

   Explanation: The relation between Quartile Deviation (QD), Mean Absolute Deviation (MAD), and Standard Deviation (SD) is an approximation and is generally acceptable. This approximated ratio is often used for comparative studies. Therefore, this statement is correct.

4. Statement 4: If X and Y are independent, then \((X-Y) \sim N(\mu_x - \mu_y, \sigma^2_x + \sigma^2_y)\).

   Explanation: If \(X\) and \(Y\) are independent and normally distributed, then the difference \((X-Y)\) is also normally distributed with mean \((\mu_x - \mu_y)\) and variance \((\sigma_x^2 + \sigma_y^2)\). This statement is correct.

Therefore, the incorrect statements are 1 and 2.