Transformation of Random Variable

In summary, the definition of X being a random variable distributed uniformly in [0, Y], where Y is geometric with mean alpha, is valid for uniform distribution. The pdf of the transformation Y-X can be found by taking the derivative of the defined probability function. It is unclear if this fully answers the question about the validity of the definition.
  • #1
hemanth
9
0
If X is a random variable distributed uniformly in [0, Y], where Y is geometric with mean alpha.
i) Is this definition valid for uniform distribution ?
ii) If it is valid, what is the pdf of the transformation Y-X?
 
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  • #2
hemanth said:
If X is a random variable distributed uniformly in [0, Y], where Y is geometric with mean alpha.
i) Is this definition valid for uniform distribution ?
ii) If it is valid, what is the pdf of the transformation Y-X?

If Y is geomtrically distributed with parameter p, that means that... $$P \{ Y = n\} = p\ (1-p)^{n}\ (1)$$

... and...

$$E \{Y\} = \sum_{n=0}^{\infty} n\ p\ (1-p)^{n} = \frac{1-p}{p}\ (2)$$

If X is uniformely distributed in [0,Y], then is...

$$ P \{Y < x \} = p + p\ \sum_{n=1}^{\infty} \varphi_{n} (x)\ (1-p)^{n}\ (3)$$

... where...

$$\varphi_{n} (x)=\begin{cases}\ 0 & \text{if}\ x < 0 \\ \frac{x}{n} &\text{if}\ 0 \le x \le n \\ 1 &\text{if} x> n \end{cases}\ (4)
$$

Of course the p.d.f. of Y is the derivative of (3). I don't know if all that answers the question i) ... Kind regards $\chi$ $\sigma$
 

FAQ: Transformation of Random Variable

What is a random variable?

A random variable is a mathematical concept used to model and describe the possible outcomes of a random process. It can take on different values with different probabilities, and is often represented by a letter such as X or Y.

What does it mean to transform a random variable?

To transform a random variable means to apply a mathematical function to its values, resulting in a new random variable. This can help simplify the data or make it more useful for analysis.

What is the purpose of transforming a random variable?

The main purpose of transforming a random variable is to make the data easier to understand and analyze. It can also help to meet certain assumptions or conditions for statistical analysis.

What are some common transformations used for random variables?

Some common transformations used for random variables include logarithmic, exponential, and power transformations. Other transformations may also be used depending on the data and the specific analysis being performed.

Can transforming a random variable change its distribution?

Yes, transforming a random variable can change its distribution. For example, a log transformation can change a skewed distribution into a more symmetrical one. However, it is important to note that the transformed variable is still considered a random variable and can still be used in statistical analysis.

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