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?
True, True, False
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.
Why is c) false? Can you give an example?
a) Is 2X-4 always normal?
Well, if X is a normal random variable, then 2X is simply stretching or compressing the distribution along the x-axis. However, subtracting 4 is like giving the distribution a ride on the "Depression Express". So, no, 2X-4 is not always normal. It's got a touch of the blues.
b) Is 3X-4Y always normal?
Ah, the whimsical combination of X and Y! When you multiply X by 3, it's like putting it on steroids. And when you multiply Y by -4, it's like giving it a spin on a rollercoaster. These two random variables have their unique quirks, but when mixed together, the result is not necessarily normal. So, no, 3X-4Y is not always normal. It's a wild ride!
c) Is X^2+Y always normal?
The combination of X squared and Y is like mixing oil with water. They just don't blend well together. Squaring X is like turning up the volume to 11, while Y is there, minding its own business. These two variables, when added together, create a bit of a mess. So, no, X^2+Y is not always normal. It's a chaotic concoction!
To determine whether the given random variables are always normal, we should consider the properties of normal random variables and apply basic rules of algebra.
A normal random variable with mean μ and variance σ^2 is defined as the sum of a constant and a multiple of a standard normal random variable, i.e., Z = a + b*W, where Z is the normal random variable, a is the constant, b is the multiplier, and W is a standard normal random variable.
a) Is 2X - 4 always normal?
To determine if 2X - 4 is always normal, we need to express it in the form a + b*W. Using algebraic manipulation, we can rewrite 2X - 4 as 2(X - 2). Since (X - 2) is still a normal random variable (due to the linearity of the normal distribution), we can conclude that 2X - 4 is also normal.
b) Is 3X - 4Y always normal?
Similar to the previous question, we need to determine if 3X - 4Y can be expressed in the form a + b*W. Since X and Y are independent normal random variables, the linear combination 3X - 4Y is known as a linear combination of normal random variables. To determine whether it is normal, we need to check if every linear combination of independent normal random variables will always be normal. In this case, 3X - 4Y is not guaranteed to be normal.
c) Is X^2 + Y always normal?
Unlike the previous questions, X^2 + Y involves a non-linear operation (squaring). The sum of two independent normal random variables is not always guaranteed to be normal, especially when a non-linear function is applied. Therefore, X^2 + Y is not guaranteed to be normal.
In conclusion, 2X - 4 is always normal, 3X - 4Y is not guaranteed to be normal, and X^2 + Y is not guaranteed to be normal.