Square root transformation of Poisson process. How Var[P(sqrt(lambda))] ~ 1/4?

I am working on Kaggle Neural data challenge.

A number of spikes given a stimulus are Poisson distributed as

$$Y_i \sim P(\lambda_i)$$

The mean and variance of any Poisson process is given as

$$E[P(\lambda_i)] = Var[P(\lambda_i)] = \lambda_i$$

In order to normalize the data, the square root transformation is applied on mean and variance as,

$$E[\sqrt{P(\lambda)}] \approx \sqrt{\lambda}$$

$$Var[\sqrt{P(\lambda)}] \approx \frac{1}{4}$$

I do not understand how the variance becomes constant by square root transformation?

Update: Received an answer at Statistics

• Hi, please cite the textbook or math page which makes that claim. In addition, I would recommend you ask this question in Math.SE, since it's purely an arithmetic question, regardless of the actual data source. – Carl Witthoft Jan 18 at 13:39
• In fact, the answer is already there! math.stackexchange.com/questions/1536459/… . Will folks with sufficient points please close this and reference that page? – Carl Witthoft Jan 18 at 13:41
• @CarlWitthoft, Thanks for the link. I posted the question in Statistics and it is answered there. Please close this question in reference to stats.stackexchange.com/questions/505298/… – sarannns Jan 18 at 19:11