Let F(Z) be the cumulative density function of the standard normal variate Z, then which of the following are correct?

(A) $F(Z) =\frac{1}{\sqrt{2\pi}}\int\limits_{-∞}^Ze^{-\frac{Z^2}{2}}dz, -∞ <Z< ∞$
(B) $F(-Z) = 1- F(Z)$
(C) $F(0) = 0$
(D) $F(∞) = 1$

Choose the correct answer from the options given below:

(A), (B) and (D) only

The correct answer is Option (1) → (A), (B) and (D) only

Let $F(Z)$ denote the cumulative distribution function of the standard normal variate $Z$

(A) $F(Z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{Z} e^{-z^2/2}\,dz$

This is the definition of the cumulative distribution function of the standard normal distribution

True

(B) $F(-Z)=1-F(Z)$

The standard normal distribution is symmetric about zero, hence this property holds

True

(C) $F(0)=0$

For a standard normal distribution, $F(0)=\frac{1}{2}$

False

(D) $F(\infty)=1$

The total probability under the standard normal curve is $1$, hence the cumulative probability tends to $1$ as $Z$ tends to infinity

True

The correct options are (A), (B) and (D).