If a random variable X has a normal distribution with mean µ and variance σ2 then: (d) (X−µ)2 σ2 has a chi-squared distribution with n degrees of freedom . Therefore , (d) (X−µ)2 σ2 has a chi-squared distribution with n degrees of freedom is correct .
Let's analyze each option:
(a) X takes positive values only:
This statement is not necessarily true. A normal distribution can take both positive and negative values.
The normal distribution is symmetric around its mean, and its range extends to negative and positive infinity.
(b) X−µ/σ has a standard normal distribution:
This statement is correct. If you standardize a normal random variable X by subtracting its mean (µ) and dividing by its standard deviation (σ), you get a standard normal random variable Z.
This is denoted as (X−µ)/σ, and it follows a standard normal distribution.
(c) X^2 has a chi-squared distribution with 1 degree of freedom:
This statement is not accurate. If X has a normal distribution, then X^2 does not follow a chi-squared distribution with 1 degree of freedom.
The chi-squared distribution with 1 degree of freedom is obtained from the square of a standard normal random variable.
(d) (X−µ)^2/σ^2 has a chi-squared distribution with n degrees of freedom:
This statement is incorrect. The expression (X−µ)^2/σ^2 does not have a chi-squared distribution with n degrees of freedom.
It actually follows a chi-squared distribution with 1 degree of freedom.
(e) None of the above:
The correct answer is (e) because only option (b) is true.
The other statements are either false or not applicable in the context of a normal distribution.