1. ### Distribution invariant under transformation

On the page on relationships among probability distributions on Wikipedia there is a line saying that "some distributions are invariant under a specific transformation". Could anyone explain what it means please.
2. ### data transformation

If Y=log(X) is normally distributed, then X is lognormally distributed. Does this mean that if I plot the probability density function of Y, the x-axis is log(x)?
3. ### Generating a uniform function from an exponential function

Thank you GretaGarbo and Dason. Now I can describe what the transform does with words and graphs. However, I am still confused when it comes to the formulas, especially the x's. The formula for the CDF is and putting X into this gives . So, if I have a dataset (x1, x2,...,x50) and I put them...
4. ### Generating a uniform function from an exponential function

I'm sorry I don't understand. I know you are right but I don't get why we get a uniform. If I put x's were the x is in the CDF, shouldn't I get a CDF? (I'm sorry if I am not very clear, I'm very confused and I really appreciate your help!)
5. ### Generating a uniform function from an exponential function

If I have some values x. From these I estimate the PDF of an exponential function f(x)= . Let's say the estimated lambda equals 50. The CDF is . How do I generate a standard uniform distribution from this?
6. ### data tranformation

Sorry for removing the question. I have posted it again.
7. ### data tranformation

How do I interpret this notation? If QN(p) is the normal quantile function, then F(Y)= QN[G(Y)] follows a normal for any G ∼ G(Y)
8. ### Degrees of freedom and sampling with replacement

The modified population variance (divisor n-1) can be explained by degrees of freedom. The mean is fixed so specifying the value of any n-1 determines the remaining value. I understand how this applies to sampling without replacement. Why doesn't this apply to sampling with replacement from a...
9. ### Inverse of CDF

I am so thankful! That is exactly what I needed to get my head around the notation. Thank you very much!!
10. ### Inverse of CDF

I am struggling to understand the second formula. I understand the first formula. I don't understand why F(x) is greater than or equal to y in the second formula. Shouldn't the sign be similar to the first formula?

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12. ### MMSE vs LSE

title without abbreviations: minimum squared error, ordinary least square, minimum mean square, least square, mean squared I am trying to understand methods for point estimation and their hierarchy in an applied perspective I understand the concept of least squared error. I understand that...
13. ### Defintion of a random variable

Thank you Dason and hlsmith!
14. ### Interpretation of inverse CDF

For the PDF above what is the inverse CDF of 0 and 0.75, i.e. Q(0) and Q(0.75)?
15. ### Defintion of a random variable

I have read two defintions for a random variable. 1. It is a function from the set of all posisible outcomes to the real line. For example Y("male")=0 2. A variable where known probabilities are associated with sample outcomes. Which one is correct? Defintion 1 does not include a probability...