Properties of Normal Distribution

Question

Describe if the following statement is true or false. Explain why.

1. There are a limited number of normal distributions.
2. There is only one standard normal distribution.
3. The mean of a normal distribution is always 0.
4. The mean of a standard normal distribution is always 0.
5. The standard deviation of the standard normal distribution is always 0.

Answer 1

There are a limited number of normal distributions.

False. A normal distribution is parametrized thru its mean and standard deviation, where the mean is a real number and the standard deviation is a positive real number. That said, we could model infinitely many number of normal distributions

There is only one standard normal distribution.

True. It is a normal distribution with mean = 0 and standard deviation = 1.

The mean of a normal distribution is always 0.

False. This only applies to the standard one.

The mean of a standard normal distribution is always 0.

True. The standard normal distribution is a normal distribution mean = 0 and standard deviation = 1.

The standard deviation of the standard normal distribution is always 0.

False. The standard normal distribution is a normal distribution mean = 0 and standard deviation = 1.

Answer 2

There are infinite number of normal distributions since both mu and sigma can take any real values.

There is only one standard normal, corresponding to mu=0, and sigma= 1.

For the last three only the standard normal is required to be N(0,1).