# a) Which of the following properties distinguishes the standard normal distribution from

Question
Normal distributions

a) Which of the following properties distinguishes the standard normal distribution from other normal distributions?
-The mean is located at the center of the distribution.
-The total area under the curve is equal to 1.00.
-The curve is continuous.
-The mean is 0 and the standard deviation is 1.
b) Find the probability $$\displaystyle{P}{\left({z}{<}-{0.51}\right)}$$ using the standard normal distribution.
c) Find the probability $$\displaystyle{P}{\left({z}{>}-{0.59}\right)}$$ using the standard normal distribution.

2021-01-14

Step 1
Standard Normal Distribution is symmetric continuous distribution with mean 0 and variance 1.
-The mean is located at the center of the distribution.
This is True because here the center of the distribution is 0 and the mean is also 0.
-The total area under the curve is equal to 1.00.
This is True because area under curve of any distribution will be always 1
-The curve is continuous.
This is True because it takes values on continuous domain
-The mean is 0 and the standard deviation is 1.
This is True by the definition of Standard Normal Distribution.
Step 2
b. $$P(Z< -0.51) = \Phi(-0.51) = 0.305026 [ \text{From the table of Standard Normal Distribution} , \Phi\ \text{is the CDF of Standard Normal Distribution} ]$$
c. $$P(z > - 0.59) = 1 - \Phi(-0.59) = 1 - 0.277595 = 0.722405 [ \text{From the table of Standard Normal Distribution} , \Phi\ \text{is the CDF of Standard Normal Distribution} ]$$

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