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I have 2 independent RVs $s$ and $N$ with distribution as below:

$\begin{array} { c } { f _ { s } ( s ) = \frac { 1 } { \sqrt { 2 \pi \sigma ^ { 2 } } } e ^ { - s ^ { 2 } / 2 \sigma ^ { 2 } } } \\ { \text { and } } \\ { f _ { N } ( n ) = \left\{ \begin{array} { c l } { b e ^ { - b n } } & { n \geq 0 } \\ { 0 } & { \text { otherwise } } \end{array} \right. } \end{array}$

Now, I want to compute the distribution of $A S ^ { 2 } + n$?

Can anyone let me know how to solve for this?

My approach so far:

Condition on $s$ and evaluate the density of $A S ^ { 2 } + n$. Then multiply with density of $s$ and integrate out $s$.

But, I am not able to simplify it considerably. Can someone show how to solve this?

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Firstly, you have a random variable $$S\sim N(0,\sigma^{2})$$

and a random variable $$N\sim \text{Exp}(b)$$

Now, we can say that $$\frac{S^{2}}{\sigma^{2}}\sim\chi^{2}_{1}\Rightarrow S^{2}=\sigma^{2}\chi^{2}_{1}\sim\text{Gamma}(1/2,2\sigma^{2})$$

So, $$AS^{2}\sim\text{Gamma}(1/2,2A\sigma^{2})$$

Now, we can note that the Exponential distribution is a special case of the Gamma distribution. Specifically, $$N\sim\text{Exp}(b)\equiv\text{Gamma}(1,b)$$

Now, let's define $$Z=AS^{2}+N$$

If $b=2A\sigma^{2}$ then finishing the problem is simple. However, if this is not the case then you have the sum of Gamma random variables with different scale parameters, which is a more complicated situation. See Generic sum of Gamma random variables for details.

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