The curve $y = f(x)$ is normal probability curve, then which of the following statements are correct?

(A) mean, median and mode of the distribution coincide.

(B) the area bounded by the curve $y = f(x)$ and $x$-axis is one unit.

(C) The curve is symmetrical about the line $x = \mu$, where $\mu$ is the mean.

(D) $y$-axis is an asymptote to the curve.

Choose the correct answer from the options given below:

A normal probability curve is a symmetric, bell-shaped curve that represents a normal distribution.

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For statement (A): In a normal distribution, the curve is perfectly symmetric. The highest point of the curve (mode) occurs at the center. Due to the symmetry, the median also lies at this center point, and the mean equals the median for a symmetric distribution.

Therefore, mean = median = mode = $\mu$

Statement (A) is correct.

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For statement (B): A probability distribution must account for all possible outcomes. The total probability of all outcomes must equal 1.

The area under the probability density curve represents total probability.

Area under the curve $= 1$

Statement (B) is correct.

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For statement (C): The normal curve is symmetric about a vertical line passing through its mean $\mu$.

For any point at distance $d$ to the left of $\mu$, there exists a corresponding point at distance $d$ to the right of $\mu$ with the same height.

The curve is symmetric about the line $x = \mu$

Statement (C) is correct.

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For statement (D): As $x \to \pm\infty$, the curve approaches the $x$-axis (horizontal asymptote).

The $y$-axis (line $x = 0$) has no special significance for a normal curve. The curve can intersect the $y$-axis at the point $(0, f(0))$ depending on the value of $\mu$.

The $y$-axis is not an asymptote.

Statement (D) is incorrect.

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Statements (A), (B), and (C) are correct.