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:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
The normal probability curve is bell-shaped, with a single peak at the mean \(\mu\). Its two tails come closer and closer to the x-axis on both sides but never touch it, and, being a probability curve, it encloses a total area of 1 with the x-axis.
So the curve is symmetric about \(x = \mu\), its peak is at \(x = \mu\), the total area under it is 1, and its asymptote is the x-axis.
Checking the statements
(A) mean, median and mode of the distribution coincide. — Correct. Symmetry about \(\mu\) and the single peak at \(\mu\) make all three equal to \(\mu\).
(B) the area bounded by the curve \(y = f(x)\) and x-axis is one unit. — Correct. The total area is the total probability, 1.
(C) The curve is symmetrical about the line \(x = \mu\), where \(\mu\) is the mean. — Correct. The curve is a mirror image about \(x = \mu\).
(D) y-axis is an asymptote to the curve. — Incorrect. The asymptote is the x-axis; the curve meets the y-axis at a finite height.