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rvkhatri
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How to transform a random variable CDF to a standard normal
Given F(x) = 1- exp (-sqrt x), for x greater that 0
Thanks.
Given F(x) = 1- exp (-sqrt x), for x greater that 0
Thanks.
rvkhatri said:How to transform a random variable CDF to a standard normal
Given F(x) = 1- exp (-sqrt x), for x greater that 0
Thanks.
I like Serena said:Welcome to MHB, rvkhatri! :)
What do you mean by "standard normal"?
Do you mean the PDF?
Or perhaps an equivalent normal distribution?
rvkhatri said:I meant standard normal distribution i.e. mean = 0, sigma = 1
My class notes say,
if F(x) = 1- exp (-x), there could be one-to-one transformation to a standard normal distribution. But I am not able to get a start on this.
I like Serena said:Suppose the transformation is given by $y=g(x)$.
That is, if X is distributed according to your exponential F(X), then we will have g(X) ~ N(0,1).
Let $\Phi(y)$ be the CDF of the standard normal distribution.
Then the transformation $g$ needs to be such that the standard normal cumulative probability up to y must be the same as the exponential cumulative probability up to x.
As a formula:
$$\Phi(y) = F(x)$$
In other words:
$$y = g(x) = \Phi^{-1}(F(x))$$
rvkhatri said:My class note gives me exactly this formula for trasformation.
Now how do we get value of y in terms of x.
The CDF of a random variable can be transformed into a standard normal distribution by using the inverse of the cumulative distribution function (CDF) of a standard normal distribution. This transformation is also known as the Box-Cox transformation.
The purpose of transforming a CDF to a standard normal distribution is to simplify and standardize the data. This allows for easier comparison and analysis of different sets of data, as well as making it easier to apply statistical tests and models.
The transformation formula for F(x)=1-exp(-sqrt x) is derived using the Box-Cox transformation method, which involves taking the natural log of the original data and then applying a power transformation. In this case, the power transformation is taking the square root of the original data.
Using a standard normal distribution has several advantages, including allowing for easier calculation and interpretation of statistical measures such as mean, standard deviation, and correlation. It also allows for the use of many common statistical tests and models, such as the t-test and ANOVA.
No, not all CDFs can be transformed into a standard normal distribution. The CDF must have a known inverse function and the data must be continuous and normally distributed for the transformation to be valid. Additionally, some datasets may require a different type of transformation to achieve normality.