Any normal distribution has a variance of one and a mean zero
Select one:
- True
- False

Question

Any normal distribution has a variance of one and a mean zero  Select one:  - True  - False

Blanche McClain · Accepted Answer

Step 1
A normal distribution is a continuous type probability distribution. It is also known as a Gaussian distribution. It is a two parameter distribution with mean denoted by

μ

and standard deviation denoted by

σ

. The normal distribution is of great importance in probability distribution and statistics because of the concepts like, central limit theorem and normality assumptions in various hypothesis tests.
The probability density function of a normal distribution is given by:

\Rightarrow f (x) = \frac{1}{\sqrt{2 π σ^{2}}} e^{- \frac{1}{2} {(\frac{x - μ}{σ})}^{2}}

Where, f(x) is the probability density function of a random variable x.
Step 2
The standard normal distribution is the transformation of a normal distribution. If we subtract a mean from a normally distributed random variable x and divide it by the standard deviation, then we get the standard normal variable. Z score is nothing but the standard normal variate. Z- Scores are given by the following formula:

\Rightarrow Z - S c or e = \frac{X - \overset{―}{X}}{S . D . (X)}

Where, X is a normally distributed random variable and S.D.(X) is the standard deviation of X.
Therefore, only the standard normal distribution has a variance of 1 and a mean 0.
The value of mean and variance of normal distributions may vary according to the data values in the distribution.
Therefore, the above statement is false.

Any normal distribution has a variance of one and a

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