MHB Transform Random Var CDF to Standard Normal: F(x)=1-exp(-sqrt x)

[rvkhatri]

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]

How to transform a random variable CDF to a standard normal
Given F(x) = 1- exp (-sqrt x), for x greater that 0

Thanks.

Welcome to MHB, rvkhatri! :)

What do you mean by "standard normal"?

Do you mean the PDF?
Or perhaps an equivalent normal distribution?

 

[rvkhatri]

Welcome to MHB, rvkhatri! :)

What do you mean by "standard normal"?

Do you mean the PDF?
Or perhaps an equivalent normal distribution?

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]

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.

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]

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:
$$y = g(x) = \Phi^{-1}(F(x))$$

My class note gives me exactly this formula for trasformation.

Now how do we get value of y in terms of x.

 

[I like Serena]

My class note gives me exactly this formula for trasformation.

Now how do we get value of y in terms of x.

We already have.

You're probably thinking of rewriting it into an expression using only standard functions.
But I'm afraid we can't.
The function $\Phi(x)$ cannot be expressed as a finite combination of standard functions (this has been proven mathematically).
As a result $\Phi^{-1}(1-e^{-x})$ cannot be expressed in such a form.

The expression we have is as simple as it gets.