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:
Answer & explanation
Correct answer: option 1
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).