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In my research I am currently conducting a monte carlo simulation, in which I end up computing the sum of say $X$ and $Y$, where $X \sim N(0,1)$ and $Y = X^2$. That is, I sum up a standard normal distributed random variable and the square of this random variable. Now, this is not so important for the simulation itself, but I am wondering about the distribution and the properties (mean, variance) of the random variable $Z$, if $Z = X + Y$.

What I have come up with so far is that $Y$ is actually just a $\mathcal{X^2}$ distributed random variable with one degree of freedom.

So essentially, the question boils down to: What are the properties of a random variable that is the sum of a standard normal and a chi-squared distributed variable?

Max
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    Complete the square and you will see that you see that $Z$ follows location-shifted non-central chi-square distribution, see https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution – Jarle Tufto Feb 17 '23 at 09:32
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    If $X$ is a measurement (of, e.g., length), adding $X$ to $X^2$ is not particularly coherent as it means adding meters and squared meters. – Xi'an Feb 17 '23 at 10:39
  • @Jarle Tufto What do you mean with "complete the square"? You mean actually squaring both parts of $Z$?

    I know that this will results in some form of chi-squared distribution, but I am particularly interested in the case where we combine a standard normal distributed random variable with a chi squared distributed random variable.

     – Max Feb 17 '23 at 14:35
  • @Xi'an We are just doing a simulation, so $X$ has no real meaning to be honest. But you can think of it as we are trying to mimic a Data Generating Process in which the Variable X not only impacts our target in a linear way, but also in a non-linear (quadratic) fashion. – Max Feb 17 '23 at 14:36
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    @Max See https://en.wikipedia.org/wiki/Completing_the_square. In this case, $Z=X+X^2=(X+1/2)^2 - 1/4$. It follows that $(X+1/2)^2$ has a non-central chi-square distribution. – Jarle Tufto Feb 17 '23 at 14:37
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    Please search our site. – whuber Feb 17 '23 at 15:09

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