Sequence of normalized random variables

In summary: Suppose \(X_{2k}\sim N(1,1) \) and \( X_{2k-1} \sim U(0,2),\ k=1, 2, ..\) Now does the sequence \(\{X_i\}\) converge (in whatever sense ..) ?If the X's are normally distributed with mean 1, then the sequence \(\{X_i\}\) converges to a random variable with mean 1.
  • #1
batman3
3
0
Let X_1, X_2, ... be a sequence of random variables and define Y_i = X_i/E[X_i]. Does the sequence Y_1, Y_2, ... always convergence to a random variable with mean 1?
 
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  • #2
batman said:
Let X_1, X_2, ... be a sequence of random variables and define Y_i = X_i/E[X_i]. Does the sequence Y_1, Y_2, ... always convergence to a random variable with mean 1?

There is something missing from your statement of the problem, the \(Y_i\)s all have mean 1, but there is no reason why they should converge without some further restriction on the \(X\)s.

CB
 
  • #3
So what should be the restriction on the X_i's? In particular, can the Y_i's still convergence if the X_i's go to infinity?
 
  • #4
batman said:
So what should be the restriction on the X_i's? In particular, can the Y_i's still convergence if the X_i's go to infinity?

It would be better if you provide the context and/or further background for your question.

CB
 
  • #5
There is no context and/or further background. It was just a question that came to my mind while studying convergence of random variables. I just thought a normalized sequence would always have mean 1 in the limit and I was wondering whether there would be a general condition when it converges.
 
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  • #6
what if the X's blow up like

$P(X_n=2^n)=2^{-n}$ and $P(X_n=0)=1-2^{-n}$
 
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  • #7
batman said:
There is no context and/or further background. It was just a question that came to my mind while studying convergence of random variables. I just thought a normalized sequence would always have mean 1 in the limit and I was wondering whether there would be a general condition when it converges.

Suppose \(X_{2k}\sim N(1,1) \) and \( X_{2k-1} \sim U(0,2),\ k=1, 2, ..\) Now does the sequence \(\{X_i\}\) converge (in whatever sense ..) ?
 

FAQ: Sequence of normalized random variables

What is a sequence of normalized random variables?

A sequence of normalized random variables is a series of random variables that have been transformed to have a mean of 0 and a standard deviation of 1. This allows for easier comparison and analysis of the data.

How is a sequence of normalized random variables calculated?

To calculate a sequence of normalized random variables, the data is first standardized by subtracting the mean and dividing by the standard deviation. This results in a distribution with a mean of 0 and a standard deviation of 1. This standardized data is then used as the new values for the random variables in the sequence.

What are the benefits of using a sequence of normalized random variables?

Using a sequence of normalized random variables allows for easier comparison and analysis of data, as all variables have been transformed to have the same scale. This can also make the data more interpretable and can help identify patterns or relationships between the variables.

Can a sequence of normalized random variables be used for any type of data?

Yes, a sequence of normalized random variables can be used for any type of data, as long as the data is numeric and follows a normal distribution. If the data does not follow a normal distribution, other transformations may be necessary.

Is there a difference between a sequence of normalized random variables and a sequence of standardized random variables?

Yes, there is a difference between the two. While both involve transforming the data to have a mean of 0 and a standard deviation of 1, a sequence of normalized random variables uses the actual mean and standard deviation of the data, while a sequence of standardized random variables uses the theoretical mean and standard deviation of the underlying distribution.

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