What happens when you substract a Normal distribution with a constant?

[lee22042017]

My final goal is to find the variance and mean of S.

I have no idea at all, so I would like to find out what happens to the Normal distribution when it is substracted by a constant first.

 

[BigBeachBanana]

View attachment 31871
My final goal is to find the variance and mean of S.

I have no idea at all, so I would like to find out what happens to the Normal distribution when it is substracted by a constant first.

What you posted doesn't make sense or incomplete
What is $Y_i?$
N is the index, how can it be independent?
You defined $S=\sum_{i=1}^{N}X_i$, where does $Y_i$ come into play?
Perhaps attach the original problem.

In general,
If $X\sim N(\mu,\sigma^2)$ then,
(1) $X+c\sim N(\mu+c,\sigma^2)$
(2) $cX\sim N(c\mu,c^2\sigma^2)$

 

[lee22042017]

What you posted doesn't make sense or incomplete
What is $Y_i?$
N is the index, how can it be independent?
You defined $S=\sum_{i=1}^{N}X_i$, where does $Y_i$ come into play?
Perhaps attach the original problem.

In general,
If $X\sim N(\mu,\sigma^2)$ then,
(1) $X+c\sim N(\mu+c,\sigma^2)$
(2) $cX\sim N(c\mu,c^2\sigma^2)$

Xi = Yi - 816 when y > 816. Thanks a lot

 

[BigBeachBanana]

Xi = Yi - 816 when y > 816. Thanks a lot

Now I understand. That's a terrible notation for piecewise function, and I think there's a typo. There shouldn't be any Y. You're modelling claim payouts with a deductible of 816, meaning the insurer pays nothing if the claim is less than 816, and pay anything over 816.
Notice Claim Count, $N\sim Poisson(\lambda=8.8)$
$$p_k = \begin{cases} 0.17 &\text{for } k=0,1 \\ c(\frac{e^{-8.8}8.8^k}{k!}) &\text{for } k=2,3,\dots \end{cases}\\$$and the Claim Size,
$$X_i= \begin{cases} 0 &\text{for } x_i\le 816.0 \\ x_i-816.0 &\text{for } x_i>816.0 \end{cases}$$