Transformations of random variables

Good luck on your exam!In summary, to work out the distribution of Y=X^2, you need to rearrange the equation and substitute values from the original pdf, while for fX(2x), you simply replace x with 2x in the original pdf. No differentiation is needed. Good luck on your exam!
  • #1
stukbv
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0
Hi, I am a bit confused.
Basically if I have a pdf, fX(x) and i want to work out the distribution of Y=X^2 for example, then this involves me letting Y=X^2, rearranging to get X in terms of Y, substituting these into all values of x in my original pdf fX, and then multipying it by whatever dx = .

But if i have a pdf fX(x) and just want to work out fX(2x) , this just means put 2x in place of all the x's and that is that. No differentiation etc.

Is this right?

Thanks a lot - I have an exam tomorrow!
 
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  • #2
Yes, that is correct. To work out the distribution of Y=X^2, you need to rearrange the equation to solve for X in terms of Y, then substitute all of the values of X from the original pdf into the new equation. Then multiply it by whatever dx =. For fX(2x), all you need to do is replace all of the x's in the original pdf with 2x, and that is it. No differentiation is required.
 

1. What is a transformation of a random variable?

A transformation of a random variable is a mathematical process that involves manipulating the values of a random variable to obtain a new random variable. This can involve adding or multiplying constants, taking logarithms or exponentials, or applying other mathematical functions to the original random variable.

2. Why are transformations of random variables important?

Transformations of random variables are important because they allow us to simplify complex probability distributions and make them easier to analyze. They also help us to better understand the relationship between different random variables and how they affect each other.

3. How do you determine the distribution of a transformed random variable?

The distribution of a transformed random variable can be determined by using the transformation function to map the original probability distribution onto the new random variable. This can be done by using the transformation formula or by using graphical methods such as the transformation plot.

4. What is the difference between a linear and non-linear transformation of a random variable?

A linear transformation of a random variable results in a new random variable that has a linear relationship with the original variable. This means that the values of the new variable can be expressed as a linear combination of the values of the original variable. Non-linear transformations, on the other hand, result in a new random variable that has a non-linear relationship with the original variable.

5. Can a transformation change the shape of a probability distribution?

Yes, a transformation can change the shape of a probability distribution. Depending on the type of transformation, the distribution can become more skewed or symmetric, have heavier or lighter tails, or have a different overall distribution shape. This is why it is important to understand the effects of different transformations on probability distributions.

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