Suppose I have two variables $X$ and $Y$ jointly drawn from a normal distribution with $\mathrm{Cov}[X,Y] = 0$. Defining $Z = X+Y$, what is the most straightforward way to compute $\mathbb{E}\left[X|Z\right]$?

I am writing \begin{align} \mathbb{E}\left[X|Z\right] & = \int_x x f_{X}(x|Z=z) \mathrm{d}x \\ & = \int_x x \frac{f_{XZ}(x,z)}{f_Z(z)} \mathrm{d}x \\ & = \int_x x \frac{f_{Y}(z-x)f_X(x)}{f_Z(z)} \mathrm{d}x, \\ \end{align}

but I am not sure if the best (and only?) way is to compute this expression using our knowledge of the pdf's. Thanks.

Suppose I have two independent normal variables $X$ and $Y$ with known mean and variance. Defining $Z = X+Y$, what is the most straightforward way to compute $\mathbb{E}\left[X|Z\right]$?

I am writing \begin{align} \mathbb{E}\left[X|Z\right] & = \int_x x f_{X}(x|Z=z) \mathrm{d}x \\ & = \int_x x \frac{f_{XZ}(x,z)}{f_Z(z)} \mathrm{d}x \\ & = \int_x x \frac{f_{Y}(z-x)f_X(x)}{f_Z(z)} \mathrm{d}x, \\ \end{align}

but I am not sure if the best (and only?) way is to compute this expression using our knowledge of the pdf's. Thanks.

Suppose I have two i.i.d. variables $X \sim N(\mu_X, \sigma^2_X)$ and $Y \sim N(\mu_Y, \sigma^2_Y)$. Defining $Z = X+Y$, we have $Z \sim N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$. What is the most straightforward way to compute $\mathbb{E}\left[X|Z\right]$?

I am writing \begin{align} \mathbb{E}\left[X|Z\right] & = \int_x x f_{X}(x|Z=z) \mathrm{d}x \\ & = \int_x x \frac{f_{XZ}(x,z)}{f_Z(z)} \mathrm{d}x \\ & = \int_x x \frac{f_{Y}(z-x)f_X(x)}{f_Z(z)} \mathrm{d}x, \\ \end{align}

but I am not sure if the best (and only?) way is to compute this expression using our knowledge of the pdf's. Thanks.

Suppose I have two variables $X$ and $Y$ jointly drawn from a normal distribution with $\mathrm{Cov}[X,Y] = 0$. Defining $Z = X+Y$, what is the most straightforward way to compute $\mathbb{E}\left[X|Z\right]$?

I am writing \begin{align} \mathbb{E}\left[X|Z\right] & = \int_x x f_{X}(x|Z=z) \mathrm{d}x \\ & = \int_x x \frac{f_{XZ}(x,z)}{f_Z(z)} \mathrm{d}x \\ & = \int_x x \frac{f_{Y}(z-x)f_X(x)}{f_Z(z)} \mathrm{d}x, \\ \end{align}

but I am not sure if the best (and only?) way is to compute this expression using our knowledge of the pdf's. Thanks.