Table of Contents
Will the product of two normally distributed random variables be normally distributed?
When is the distribution of product of two normal distributed variables near normal distribution? It is clear the product of normal distributed variables is not normal distributed. For example, if X∼N(μ1,σ21), Y∼N(μ2,σ22), then XY does not has the distribution of N(μ1μ2,μ21σ21+μ22σ21).
What happens if two independent normal random variables are combined?
What happens if two independent normal random variables are combined? Any sum or difference or independent normal random variables is also normally distributed. A binomial setting arises when we perform several independent trials of the same chance process and record the number of times a particular outcome occurs.
Is the product of two independent random variables independent?
Two random variables X and Y are independent if all events of the form “X ≤ x” and “Y ≤ y” are independent events. On the other hand, the expected value of the product of two random variables is not necessarily the product of the expected values.
Can you multiply a normal distribution?
This answer notes that if a programming language/libraries provide a procedure that returns random samples from a standard normal distribution, we can generate samples from another normal distribution with the same mean by multiplying the samples by the standard deviation σ of the desired distribution.
Is the sum of two normal random variables is still a normal random variable?
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations).
Is the sum of Gaussians Gaussian?
A Sum of Gaussian Random Variables is a Gaussian Random Variable. That the sum of two independent Gaussian random variables is Gaussian follows immediately from the fact that Gaussians are closed under multiplication (or convolution).
How do you prove that two random variables are independent?
If X and Y are two random variables and the distribution of X is not influenced by the values taken by Y, and vice versa, the two random variables are said to be independent. Mathematically, two discrete random variables are said to be independent if: P(X=x, Y=y) = P(X=x) P(Y=y), for all x,y.
Can the product of two normal random variables be another normal?
The above argument holds true irrespective of whether the product of two normal random variables is another normal random variable or not. As an added bonus, the product of two normal random variables may not be normal.
Is the product of two Gaussian PDFs always a Gaussian PDF?
A random variable product of two independent gaussian random variables is not gaussian except in some degenerate cases such as one random variable in the product being constant. A product of two gaussian PDFs is proportional to a gaussian PDF, always, trivially. Idem for the convolution of PDFs.
What is the product of two normal probability density functions?
The product of two normal (i.e., Gaussian) probability density functions is easily worked out, it’s simple algebra. Say the two random variables are x and y.
Can the product x y of independent normals be normal?
You can use moments to see that the product X Y of independent normals cannot be normal except in trivial cases. By trivial, I mean V ( X) V ( Y) = 0. Suppose that X, Y are independent normals so that X Y is normal.