<p>I am working on <a href="https://www.kaggle.com/deanp3/uw-neural-data-challenge-intro" rel="nofollow noreferrer">Kaggle Neural data challenge</a>.</p>
<p>A number of spikes given a stimulus are Poisson distributed as</p>
<p><span class="math-container">$$Y_i \sim P(\lambda_i)$$</span></p>
<p>The mean and variance of any Poisson process is given as</p>
<p><span class="math-container">$$E[P(\lambda_i)] = Var[P(\lambda_i)] = \lambda_i$$</span></p>
<p>In order to normalize the data, the square root transformation is applied on mean and variance as,</p>
<p><span class="math-container">$$E[\sqrt{P(\lambda)}] \approx \sqrt{\lambda}$$</span></p>
<p><span class="math-container">$$Var[\sqrt{P(\lambda)}] \approx \frac{1}{4}$$</span></p>
<p>I do not understand how the variance becomes constant by square root transformation?</p>
<p>Update: Received an answer at <a href="https://stats.stackexchange.com/questions/505298/square-root-transformation-of-poisson-process-how-small-var-sqrtp-lambda">Statistics</a></p>

CC BY-SA 4.0
2021-01-18T13:39:20.273Z
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
CC BY-SA 4.0
2021-01-18T13:41:16.163Z
In fact, the answer is already there!    https://math.stackexchange.com/questions/1536459/expected-value-of-square-root-of-poisson-random-variable   .   Will folks with sufficient points please close this and reference that page?
Carl Witthoft
CC BY-SA 4.0
2021-01-18T19:11:58.053Z
@CarlWitthoft, Thanks for the link. I posted the question in Statistics and it is answered there. Please close this question in reference to https://stats.stackexchange.com/questions/505298/square-root-transformation-of-poisson-process-how-small-var-sqrtp-lambda
Saran
<neuroscience><statistics>
Square root transformation of Poisson process. How Var[P(sqrt(lambda))] ~ 1/4?
Saran
# Square root transformation of Poisson process. How Var[P(sqrt(lambda))] ~ 1/4?

(Asked by: username_0 on 2021-01-17)

I am working on [Kaggle Neural data challenge](https://www.kaggle.com/deanp3/uw-neural-data-challenge-intro).

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](https://stats.stackexchange.com/questions/505298/square-root-transformation-of-poisson-process-how-small-var-sqrtp-lambda)

**Tags:** neuroscience, statistics

> **Comments:**
> **username_1**: 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.  
> **username_1**: In fact, the answer is already there!    https://math.stackexchange.com/questions/1536459/expected-value-of-square-root-of-poisson-random-variable   .   Will folks with sufficient points please close this and reference that page?  
> **username_0**: @CarlWitthoft, Thanks for the link. I posted the question in Statistics and it is answered there. Please close this question in reference to https://stats.stackexchange.com/questions/505298/square-root-transformation-of-poisson-process-how-small-var-sqrtp-lambda  

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