# Statistics: variable transformation proof?

• Nikitin
In summary, the conversation discusses the proof for the assumption of ##Y=u(X) \Leftrightarrow y=u(x)##. The solution involves using the probability distribution of ##X## and ##Y## to prove that ##P(Y = y) = P(X = w(y))##, with w being the inverse one-on-one function of u. It is noted that the one-to-one assumption is crucial for this proof to hold.
Nikitin

## Homework Statement

Ok this might be a stupid question, but:

https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-frc3/t31/q77/s720x720/10001118_10202561443653973_1625797585_o.jpg
Why is this the case? I think for all of this to be right, then the assumption of ##Y=u(X) \Leftrightarrow y=u(x)##. But how do I prove this?

## The Attempt at a Solution

An attempt: OK, assume ##X## has a probability distribution given by ##f(x)##, and ##Y=2X## has a probability distribution of ##g(x)##. Then if ##Y = u(X)##, with w being the inverse one-on-one function of u, ##P(X=x) = P(w(Y)=x) = P(Y=u(x))=P(Y=y=u(x))##.

This is as far as I got, but now I am confused. What to do?

You're almost there. Note by definition that

$$g(y) = P\{Y = y\}$$

and you must prove that

$$g(y) = f(w(y))$$

which is by definition

$$f(w(y)) = P\{X = w(y)\}$$

So you must somehow find why

$$P\{Y = y\} = P\{X=w(y)\}$$

1 person
OK, I see. Thanks for the help :)

Nikitin said:

## Homework Statement

Ok this might be a stupid question, but:

https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-frc3/t31/q77/s720x720/10001118_10202561443653973_1625797585_o.jpg
Why is this the case? I think for all of this to be right, then the assumption of ##Y=u(X) \Leftrightarrow y=u(x)##. But how do I prove this?

## The Attempt at a Solution

An attempt: OK, assume ##X## has a probability distribution given by ##f(x)##, and ##Y=2X## has a probability distribution of ##g(x)##. Then if ##Y = u(X)##, with w being the inverse one-on-one function of u, ##P(X=x) = P(w(Y)=x) = P(Y=u(x))=P(Y=y=u(x))##.

This is as far as I got, but now I am confused. What to do?

You already got the answer below, but just to clarify: the one-to-one assumption is crucial. If you had a non-monotone function, such as ##u(x) = x^2##, and if the range of ##X## included both positive and negative values, then the mapping ##Y = u(X)## might not be 1:1, and so you might need a more complicated evaluation of ##P(Y = y)##.

Last edited:
1 person

## 1. What is variable transformation in statistics?

Variable transformation in statistics refers to the process of converting data from one form to another in order to achieve a specific goal. This can involve changing the scale of measurement, such as transforming from a linear to a logarithmic scale, or transforming non-normal data into a normal distribution.

## 2. Why is variable transformation necessary in statistics?

Variable transformation is necessary in statistics for a variety of reasons. It can help to meet assumptions of statistical tests, such as normality and homogeneity of variance. It can also help to improve the interpretability of results or to reduce the influence of outliers on the data.

## 3. How do you determine if a variable needs to be transformed?

There is no definitive rule for determining if a variable needs to be transformed. However, some common indicators that a transformation may be beneficial include skewness or kurtosis in the data, non-linear relationships between variables, or unequal variances between groups. It is important to carefully consider the goals of the analysis and the assumptions of the statistical tests being used when deciding whether to transform a variable.

## 4. What are some common methods of variable transformation?

Some common methods of variable transformation include logarithmic transformations (such as log, ln, or square root), power transformations (such as square or cube), and categorical transformations (such as dummy coding or one-hot encoding). The choice of transformation will depend on the type of data and the goals of the analysis.

## 5. How do you know if a variable transformation has been successful?

The success of a variable transformation can be evaluated by examining the distribution of the transformed variable and assessing whether it now meets the assumptions of the statistical test being used. Additionally, visual inspection of graphs or plots can help to determine if the relationship between variables has been improved. It is also important to consider the impact of the transformation on the interpretation of the results and the practical significance of any changes in the data.

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