Expected value for a combination of density functions

Let $f_1,...,f_k$ density functions with continuous random variables $X_1,...,X_k$ and define

$$g(x):=\dfrac{f_1(x)+...+f_k(x)}{k}.\forall x\in\mathbb{R}$$

If $E[X_j]=j$, for $j=1,...,k$ and Y is a random variable with density $g$. Find $E[Y]$

Is it ok to do

$$E[Y]=\int_{-\infty}^{\infty}yg(y)dy=\dfrac{1}{k} (\int_{-\infty}^{\infty}x_1f_1(x_1)dx_1+...+\int_{-\infty}^{\infty}x_kf_k(x_k)dx_k)$$

$$=\dfrac{1}{k}(E[X_1]+...+E[X_k])=\dfrac{1}{k}(1+...+k)=\dfrac{k(k+1)}{2k}$$??

Thank you.

Solution 1:

Yes. It is fine. My working below doesn't change the integration variable to $x_i$.

\begin{align} E[Y] &= \int_{-\infty}^\infty yg(y) \, dy \\ &=\int_{-\infty}^\infty y\cdot \frac1k\sum_{i=1}^k f_k(y) \, dy \\ &=\frac1k \int_{-\infty}^\infty yf_k(y) \, dy\\ &=\frac1k \sum_{i=1}^k E[X_k]\\ &= \frac{k(k+1)}{2k}\\ &=\frac{k+1}2 \end{align}