Answer to Question #104085 in Statistics and Probability for Katenda

Question

Which one of the following statements is incorrect with regard to the standard normal distribution?
(1) The standard normal distribution has a median of zero.
(2) The standard normal distribution is symmetric around the mean of zero.
(3) The standard normal distribution has a standard deviation of 1.
(4) The standard normal distribution has a variance of 1.
(5) The area to the right side of the mean is 1 and the area to the left side of the mean is also 1

Answer 1

Let random variable X has standard normal distribution. It means that X has expectation 0 and standard deviation 1. Statements (3) and (4) are correct. Probability density function is symmetric around expectation 0. Statement (2) is correct. Statement (1) is also correct because median equals mean in normal distribution. Statement (5) is incorrect because p. d. f. is symmetric around 0 and $\int_{-\infty}^{\infty}p(t)dt =1$. So $\int_{-\infty}^{0}p(t)dt=\int_{0}^{\infty}p(t)dt=0.5$.