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What happens if two random variables have the same mean?
For example, consider a mixture of two normal distributions, each with σ 2 = 1, but their means are 2 and − 2, respectively. The resulting mixture will have a mean of μ = 0 and a variance of σ 2 = 5, which is the same expectation and variance as a single normal distribution N ( 0, 5):
How to find the mean of a random variable?
Let X be a random variable whose possible values x 1, x 2, x 3, …, x n occur with probabilities p 1, p 2, p 3 ,…, p n, respectively. The mean of X, denoted by The number of adults living in homes on a randomly selected city block is described by the following probability distribution.
When do you add or subtract random variables?
Make sure that the variables are independent or that it’s reasonable to assume independence, before combining variances. Even when we subtract two random variables, we still add their variances; subtracting two variables increases the overall variability in the outcomes.
Which is an example of a dependent random variable?
In Example 2, both the random variables are dependent . Thus the mean of the sum of a student’s critical reading and mathematics scores must be different from just the sum of the expected value of first RV and the second RV. But the answer says the mean is equal to the sum of the mean of the 2 RV, even though they are independent.
How to find the distribution of a random variable?
If X 1, X 2, …, X n >are mutually independent normal random variables with means μ 1, μ 2, …, μ n and variances σ 1 2, σ 2 2, ⋯, σ n 2, then the linear combination: We’ll use the moment-generating function technique to find the distribution of Y.
How to test if two variables follow the same distribution?
If you want to test whether two variables follow the same distribution, would it be a good test to simply sort both variables, and then check their correlation? If it is high (at least 0.9?), then the variables most likely come from the same distribution. With distribution here I mean “normal”, “chi-square”, “gamma” etc.
Can a continuous distribution have the same mean and variance?
Another example is multimodality: A continuous distribution with multiple modes can have the same mean and variance as a distribution with a single mode, while clearly they are not identically distributed. For example, consider a mixture of two normal distributions, each with σ 2 = 1, but their means are 2 and − 2, respectively.
What does the mean of the same distribution mean?
It means that F X ( t) = P ( X ≤ t) = P ( Y ≤ t) = F Y ( t), for all t. Then, in particular, if the mean and variance exist, then their values coincide for these variables. The functions F X ( t) = P ( X ≤ t) and F Y ( t) = P ( Y ≤ t) are termed the distribution functions of the variables X and Y, respectively. See
Can you show that x 1 and x 2 have the same distribution?
I need to show that their joint distribution lies in a 1-dimensional space in R 2, since it is a singular normal distribution. I would like to conclude that X 1 = X 2 and thus the 1D space is just the diagonal. But can I go from ” X 1 and X 2 have the same distribution to saying X 1 = X 2?