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Let $X$ be a Poisson random variable with mean $\lambda$. Show that $var(\sqrt{x})\approx.25$

Here is my attempt:

In a previous part of the problem, I showed $E[g(x)]\approx g(\mu)+\frac{g''(\mu)\sigma^2}{2}$, where $\mu$ is the mean and $\sigma^2$ is the variance of the random variable.

Let $g(x)=\sqrt{x}$. Then $E[\sqrt{x}]\approx \sqrt{\mu}-\frac{\sigma^2}{8{\mu}^{3/2}}$

Thus, $var(\sqrt{x})=E[(\sqrt{x})^2]-(E[\sqrt{x}])^2\approx E[x]-(\sqrt{\mu}-\frac{\sigma^2}{8{\mu}^{3/2}})^2=\lambda-\mu+\frac{\sigma^2}{4{\mu}}-\frac{\sigma^4}{64{\mu}^3}$.

I was hoping something would cancel out to $.25$. Did I do something wrong or is there something I missed that allows me to simplify further?

Thank you in advance!

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  • $\begingroup$ what is $\sigma$ ? $\mu$? $\endgroup$ – Canardini Dec 6 '16 at 2:20
  • $\begingroup$ @Canardini $\sigma^2$ is the variance and $\mu$ is the mean of the random variable I used to find $E[g(x)]\approx g(\mu)+\frac{g''(\mu)\sigma^2}{2}$ $\endgroup$ – Silvia Rossi Dec 6 '16 at 2:27
  • $\begingroup$ @Canardini I was not given a distribution for that random variable $\endgroup$ – Silvia Rossi Dec 6 '16 at 2:30
  • $\begingroup$ If the r.v. is $X$ then you should replace $x$'s by $X$'s in the formulation. $\endgroup$ – A.G. Dec 6 '16 at 2:51
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You know the distribution of X which is Poisson of parameter $\lambda$, therefoire

$var(X)=\sigma^2=\lambda$

$E(X)=\mu=\lambda$

substitute that in your formula, which is correct btw,

you get that $var(\sqrt(X))=\frac{1}{4}-\frac{1}{64\lambda}$

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Remember that $\lambda=\mu=\sigma^2$ for a Poisson variable, so $\operatorname{Var}(\sqrt x)\approx \frac 1 4 - \frac {1}{64\lambda}$. So if $\lambda$ is large enough, we can say that this is about $\frac 1 4$.

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