Question

Considering a graphed pdf for a continuous normal distribution with mean μ and standard deviation σ, which of the following are true?

Select one or more:

a. If the mean is zero then the pdf of the normal distribution and standardised normal distribution will be the same

b. A distribution with a lower σ will have an increased P(X=μ)

c. A combination of adding constants to and multiplying each term in the sample space will affect the skewness of the graph

d. Multiplying each term in the sample space by a value greater than 1 will “flatten” the graph

e. Adding a constant to every term in the sample space will not change the shape of the graph

Answer 1

Considering the graph of continuous normal distribution with mean and standard deviation .

Statement a) - The standardised normal distribution has mean 0 and standard deviation 1. Hence, having only mean 0 doesn't make the normal distribution a standardised normal. Hence, the statement is False.

Statement b) - We know that the graph of pdf of normal distribution is a bell shaped curve symmetric around it's mean. Standard deviation gives the spread of the values around mean. If the is large then the values are scattered away from the mean and if the is lower then the values are more concentrated around mean. Hence, having lower will have an increased  P(X=μ). Hence, the statement is True.

Statement c) - We know the property of normal distribution that if the normally distributed variable is added and multiplied by a constant, then also it follows the normal distribution only. And irrespective of the mean and standard deviation, every normal random distribution is symmetric around it's respective mean. Hence, A combination of adding constants to and multiplying each term in the sample space will not affect the skewness of the graph, it will remain symmetric. Hence, the statement is False.

Statement d) - Let X follows normal distribution with mean and standard deviation . Let C be a constant greater than 1. Then, CX follows normal distribution with mean C and standard deviation C. We can see that as the C is greater than 1 the variance of CX has increased. As CX has increased variance the sample values will be less concentrated around the mean and hence the corresponding graph will have flattened shape. Hence, the statement is True.

Statement e) - Let A be any constant. Then A+X follows normal distribution with mean A+ and standard deviation . We can see that the mean changes, whereas the variance remains the same. Hence, the graph of pdf of A+X will have center shifted to new mean A+ but, the shape of the graph remains same as the variance doesn't change. Hence, the statement is True.

Hence, the statments (b), (d), (e) are True.

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