Let X and Y be independent normal random variables.
a) Is 2X−4 always normal?
b) Is 3X−4Y always normal?
c) Is X^2+Y always normal?
3 answers
True, True, False
Why is c) false? Can you give an example?
I apologize for the incorrect response. C) X^2 + Y can actually be normally distributed.
In order to determine if X^2 + Y is always normal, we need to consider whether the sum of a normally distributed random variable (X^2) and an independent random variable (Y) is also normally distributed.
The sum of two random variables would be normally distributed if their joint distribution is a bivariate normal distribution, and if the linear combination of the variables (in this case, X^2 + Y) follows a linear transformation of a normal distribution.
To illustrate this with an example, let's assume that X and Y are independent standard normal random variables, with X ~ N(0, 1) and Y ~ N(0, 1).
Let's calculate the distribution of X^2 + Y:
- Since X follows a standard normal distribution, X^2 follows a chi-squared distribution with 1 degree of freedom (X^2 ~ chi-squared(1)).
- Y follows a standard normal distribution.
- Since X^2 and Y are independent, we can take the sum of their respective distributions: X^2 + Y ~ chi-squared(1) + N(0, 1).
The chi-squared distribution with 1 degree of freedom is equivalent to an exponential distribution with parameter lambda = 1/2. The sum of an exponential and a normal distribution is known as a Skew-normal distribution. The Skew-normal distribution is a generalization of the normal distribution and does not necessarily follow a symmetric shape.
Therefore, X^2 + Y is not always normal, but follows a Skew-normal distribution.
In order to determine if X^2 + Y is always normal, we need to consider whether the sum of a normally distributed random variable (X^2) and an independent random variable (Y) is also normally distributed.
The sum of two random variables would be normally distributed if their joint distribution is a bivariate normal distribution, and if the linear combination of the variables (in this case, X^2 + Y) follows a linear transformation of a normal distribution.
To illustrate this with an example, let's assume that X and Y are independent standard normal random variables, with X ~ N(0, 1) and Y ~ N(0, 1).
Let's calculate the distribution of X^2 + Y:
- Since X follows a standard normal distribution, X^2 follows a chi-squared distribution with 1 degree of freedom (X^2 ~ chi-squared(1)).
- Y follows a standard normal distribution.
- Since X^2 and Y are independent, we can take the sum of their respective distributions: X^2 + Y ~ chi-squared(1) + N(0, 1).
The chi-squared distribution with 1 degree of freedom is equivalent to an exponential distribution with parameter lambda = 1/2. The sum of an exponential and a normal distribution is known as a Skew-normal distribution. The Skew-normal distribution is a generalization of the normal distribution and does not necessarily follow a symmetric shape.
Therefore, X^2 + Y is not always normal, but follows a Skew-normal distribution.
5 answers